text:
Movie Making Manual/Visual Effects

''Note that this page covers Visual Effects and not Special Effects. SFX are things like pyrotechnics, rain and snow. Visual Effects (VFX) are the optical tricks that are used, including projection, green-screen, miniatures, etc...''

Visual Effects have always existed; they somewhat predated Cinema. Cinema itself, based on the illusion of movement, is an Effect. Every second of projection is an illusion of movement, a lie told by the director to the audience, a mandatory preface that brings us to the practical aspects of visual effects. It is not possible to ignore the revolution in the field brought by the employment of digital software. The digital tools today replace a lot of arts in the past based just on human skills.

Visual Effects are at the core of the movie making process.
If you cannot get a special effect in the camera, then you must create the visual effect in post production. With major motion pictures, each shot can cost thousands of dollars. But for low budget film makers, there are cheaper alternatives.

Do you want to start doing amazing special effects with your small DV Camcorder? Then you have at least two options: Matte Paintings and Budget Green Screen Shooting.

Budget Green Screen Shooting
Green screen is one way of adding beautiful backgrounds to live action shots. Probably the most extreme example of this is Robert Rodrigues' Shark Boy and Lava Girl. But even if you are shooting with only a DV camcorder, you can do inexpensive green screen shots.

One can cut the cost of the shot substantially by doing your green screen shooting outdoors using a thermonuclear device (the sun) which is free (on sunny days). This means that all you have to hire is the green screen itself - none of the finely calibrated even lighting that's normally so essential for the computer to get a good key on the other end.

As an alternative to purchase or rental, one can manufacture a green screen. Commercial lighting supply houses sell paints specially manufactured for the purpose, but it is probably possible to get by with an ordinary house paint chosen carefully to be "green enough" for the computer to pull a key.

The most common difficulties with green-screen are:

 * getting even lighting on the screen.
 * getting the lighting on the foreground to match the lighting of the (separately shot) background.

By shooting outside, with a diffusion frame hung over the shot you will get naturally even sun-light all over the screen and actor. This will also match the daylight conditions of the background that later replaces the green. If your intended background is not going to be normal daylight, or if you get a lot of cloud movement, this may not work for you.

Make sure your foreground actor is a good distance from the screen, so you don't get even a hint of green light reflected back onto him/her, as this will create problems later - but with your light source being the sun (overhead), you're less likely to get green spill than if the lights were hitting the screen from the front, as they probably would in a studio.

Front Projection
Front Projection (often abbreviated FP) is a technique which can achieve the same result as green screen, but "in camera," that is to say, the composition of subject and background is complete as the combined image is acquired by the camera.

The technique uses a beamsplitter located in front of the camera in such a way as to completely fill the camera's field of view, and oriented at a precise 45 degrees to the camera's shooting axis. (This rotation from the normal may be left-right or up-down.) The other components to the system are a transparency projector (still or motion picture), and a special retro-reflective lenticular screen positioned behind the action.

A beamsplitter acts as both mirror and window, reflecting a portion of the incident light, and reflecting another portion. Beamsplitters are chosen for a specific application based on the ratio of reflectance to transmission. Common types are 50R/50T (50% reflectance, 50% transmission) and 70R/30T.

The beamsplitters used in Front Projection cinematography are of the plate type, simply a piece of plate glass with a special coating on one side designed to reduce the amount of light absorbed by the beamsplitter, and consequently neither reflected nor transmitted. The coated side faces the action, and is referred to as the "front" surface. The purpose of this coating is to reduce the loss of light due to absorption by the beamsplitter, which serves only to heat the glass.

A retroreflective screen is set behind the actors and other set pieces. This screen is not just a typical diffusive projection screen, which disperses light evenly so that a large audience composed of people sitting at many different angles to the screen sees a uniformly bright image. Instead, the retroreflective screen tends to send light right back where it came from.

The classic material for retroreflective FP screens is a made by 3M, and sold under the trademarked name of "Scotchlite". Scotchlite is used in signmaking and conspicuity applications (night time motor vehicle safety visibility). It is available from commercial signmaking supply houses.

Retroreflection in Scotchlite is achieved using millions of microscopic glass beads suspended in a transparent substrate bound to opaque vinyl sheeting. It is available in rolls of up to four feet in width.

While constructing the large (40 feet by 100 feet) screen of Scotchlite for the film 2001: A Space Odyssey, director Stanley Kubrick and special effects supervisor Tom Howard initially laid strips of Scotchlite side by side, but found that variations in manufacturing made the seams between adjacent strips glaringly obvious in the final product. Their solution involved tearing the Scotchlite into irregular overlapping pieces, minimizing the occurrence of variations of retroreflectivity large and regular enough to be discernible to the audience. Still, as Martin Hart has observed, careful examination of the FP scenes of 2001 reveal flaws introduced by variations in retroreflectivity between adjacent random patches.

A more sophisticated solution was presented in an SMTPE paper: a review of this paper will be presented in a near future version of this article.

Having discussed the nature of the physical components used in Front Projection, we turn to the preparation and arrangement of these components in a working FP system.

A still or motion picture transparency projector containing the desired background image, or "plate," is placed so that the projection axis is perpendicular to the camera's shooting axis, meeting at the place where the camera's shooting axis touches the front surface of the beamsplitter. (Thus the beamsplitter's orientation is 45 degrees to both camera and projector.)

When the projector is operating, the background plate is projected onto the front surface of the beamsplitter. A portion of the image is transmitted through the beamsplitter. In ordinary applications, the transmitted part of the image is absorbed by a black surface on the side of the beamsplitter opposite the projector, to avoid stray reflections.

The portion of the image which is not transmitted or absorbed by the beamsplitter is reflected through an angle of 90 degrees, and consequently projected over the action along the camera's shooting axis, falling onto both foreground actors and objects as well as the retroreflective screen behind them.

Retroreflective materials tend to reflect light back along the path of incidence. In FP work, the background plate image is retroreflected, back toward the beamsplitter. Part of the retroreflected background image is again lost, as it is either absorbed by the beamsplitter or reflected back into the projection lens. The remainder enters the camera where it is photographed along with the action.

The only portion of the image not accounted for in the foregoing discussion is that part of the projected background matte which falls on the actors or other foreground subjects. Foreground lighting, combined with the extreme deficit in retroreflectivity of the foreground subjects in comparison to the special screen, mean that the part of the projected image which falls on the actors is so dim as to not be detectable in-camera.

Precise alignment of system components is required to make sure that foreground objects perfectly cover their own shadows, cast by the projector on the screen. This rules panning and tilting, except in the special case where the camera is mounted so that either panning, tilting, or both occurs around the rear nodal point of the camera lens: so called "nodal pans" and "nodal tilts." In addition, the beamsplitter must be large enough, and the camera close enough, so that the camera does not take the edge of the beamsplitter into view.

Examples of nodal pan-and-tilt camera work in the context of FP can be seen in the "Dawn of Man" sequence in the film 2001: A Space Odyssey (1968), particularly the watering hole scenes. (The front projection effects on 2001 were executed by Stanley Kubrick with assistance from Tom Howard.)

A change of focal length (zoom) does not present the same difficulty as do panning or tilting. The camera can zoom in or out as long as the edges of the beamsplitter (or of the projected matte image) are not in view at the widest point of the zoom. In 2001, Kubrick also used large set pieces at either end of some FP shots in such a way as to hide the edges of his already-gigantic retroreflective screen.

In fact, a special and inventive application of zooming was used by Zoran Perisic, who worked as a rostrum or animation stand cameraman on 2001, to enhance the FP process for the film Superman: The Movie (1978). Electronically controlled motorized zoom lenses are placed on both camera and projector, and synchronized with one another so that both lenses zoom together and at the same focal length at all times. This means that the background image will not change its apparent size when the camera zooms in, as the projector simultaneously projects a reduced image. In Persic's phrase, the projector zooms and the camera zooms to "embrace" the smaller image. However, the zoom causes foreground objects to appear to rush toward or away from the camera. The combination of the "static" background and the "moving" foreground enabled the visually effective flying scenes which helped to make the film a success.

To enhance this effect still further, the use of FP in Superman introduced two other innovations: use of travelling mattes (using a motion picture projector instead of a still transparency projector, in order to project a moving background); and the mounting of the entire front projection rig (camera, projector, and beamsplitter) on a large motion-controlled robotic-arm with six degrees of freedom, and using a massive curved screen.

In rear projection process photography the projector's shutter must be synchronized with the camera's using mechanical or electronic means, in order to avoid background flicker.

The motion-controlled front-projection mount was a masterpiece of engineering for 1978, and used an early microprocessor for control. Every aspect of the rig's operation and motion could be recorded to computer tape for later automatic playback, causing the rig to move and operate exactly as trained.

Matte Paintings
The oldest and probably the most underrated visual effect is matte paintings. We see these all the time but because they look so natural, we don't notice.

Originally, matte paintings was done on glass that stood directly in front of the camera. To do matte painting, a partial set is created which is only as big as the actors and only extends to where the actors will perform. The rest of the movie set is empty space (or something that you don't want to be seen in the movie.) Except for a tiny spot of the glass which is clear, the rest of the movie set is painting on the glass. This allows you to add more scenery buildings as paintings. As long as the actors can be seen through the clear space of the glass, you cannot tell that the actors are not apart of the painted movie set.

Today, matte paintings are done with both paint and with CG (computer generated visual effects). Rather than filmed through glass, the actors are filmed normally and later composited into the matte painting. Therefore, now the distinction between matte paintings and computer generated visual effects is blurred. If the actors are filmed on a partial movie set (without any green screen, etc.) then the effect is a matte painting... even if you use computer generated effects to get the effect.

=3D Animation for Visual Effects=

When you start looking at the possibility of using 3D computer generated effects, you need understand the different types of 3D animation.

 * 1. General Purpose Animation
 * Programs such as Blender, LightWave, Maya, and 3D Studio Max are general purpose animation programs. They are very powerful, expensive (except Blender, free), have steep learning curves and are used on most high end effects movies.

 * 2. Special Purpose Animation Programs
 * Program such as Vue, Bryce, Poser, and DAZ Studio are designed for a specific purpose. Vue and Bryce are designed to create realistic scenery from nature.  Poser and DAZ Studio are designed to work with special computer models called Poser figures or Digital Puppets.  Some of these programs are even free such as DAZ Studio, Blender and Bryce.

 * 3. Special software Plug-ins
 * Software modules such as Character Studio work inside of a general purpose animation program to create a special kind of animation similar to a Special Purpose Animation Program.  LightWave, Maya, and 3D Studio Max can be greatly expanded through the use of plug-in modules.

 * 4. Support Programs
 * Programs that add in the animation but do not actually do any rendering can be extremely useful when you have a general purpose animation program which would be more awkward to use for a special task. These expand the power of  LightWave, Maya, and 3D Studio Max without making these programs too cumbersome.

3D Modeling
All elements in 3D animation must be modeled. Programs such as LightWave, Maya, XSI and 3D Studio Max come with a modeler module built in. Programs such as Vue, Bryce, Poser, and DAZ Studio do not but in the case of Poser and DAZ Studio, you can buy hundreds of figures designed for these programs. Pixologic Zbrush, and Autodesk Mudbox are especially good at modeling organic objects such as humans and creatures.

3D Animation
Animation is done in three parts, the modeling, the actual animation and the rendering. The actual animation can be:

 * 1. Keyframe animation
 * Each movement is entered into the computer system by noting the position of objects at specific frames or points in time. The movement between these key points is then calculated by the animation program based on rules set up by the animator (straight line, curved, etc.)

 * 2. Programmed animation
 * For a flock of birds, rather than record the position and movement of each bird, a computer program calculates where all the birds go and how they move.

=Compositing=

Compositing is the process of combining various elements, such as 3D imagery, live action film footage and still imagery, to create a finished shot. All visual effects that include live actors will require compositing. As mentioned above, matte paintings are no longer painted onto glass. Rather the live action is composited with matte paintings using a compositing program such as Adobe After Effects or Apple's Shake.

Software Compositing Applications Software & Hardware Compositing Systems All compositing applications provide the same basic tools and there is no hard and fast rule regarding which applications are for film and which applications are for commercials. For example, while After Effects has been used on such films as "The Aviator" it is also very widely used for broadcast and title design. Some applications do, however, come with specific tools which may prove advantageous depending on the task at hand.
 * Apple Shake (Discontinued)
 * Adobe After Effects
 * Autodesk Combustion
 * Blender
 * The Foundry Nuke
 * Eyeon Digital Fusion
 * Jahshaka
 * sony vegas pro
 * Autodesk Inferno
 * Autodesk Flame
 * Autodesk Flint
 * sony vegas pro

The primary difference between software based compositing applications and the combined software and hardware solutions is that the large Autodesk systems are significantly more expensive but provide near real-time performance at film or HD resolutions.

Compositing applications typically follow two different working paradigms.
 * Layer Based
 * Node Based

Layer based compositing applications may seem more easily approachable to the novice or beginner compositor especially those who have worked with still footage in Photoshop or The Gimp. While layer based applications can provide the same results as node based applications, node based applications are far easier to composite with when dealing with numerous elements and multiple 3D render passes. While daunting at first, node based systems may provide the compositor with greater control over the shot and easier problem solving. Even a beginning artist may benefit from learning a node based package in that she will gain a deep understanding of the exact operations which are taking place in order to create the effect.

8-Bit Graphics
Every pixel of an image used in a composite is composed of four color channels: Red, Green, Blue and Alpha. Each channel has an 8-Bit color depth resulting in a 32-Bit image. Higher color depths include 16-Bit per channel and 32-Bit per channel (float). These higher color depths allow for smoother display of color variation, for example, in gradients.

Rotoscoping
Rotoscoping, or masking, is the basis of compositing. It is the process of drawing a mask around an element in a frame or sequence of frames. The resulting image is a combination of the Red, Green and Blue color channels in addition to an Alpha channel which defines transparency.it works on frame to frame, you can not ignore a single frame which is not perfectly mask.

Tracking
Tracking is the process of matching a foreground-element's motion to that of the background. Basically you pick a decent (i.e. defined, unique, contrasty) point on your background and tell the computer to follow that point. What you get is the motion-path of your point, which you can then apply to your foreground.

Tracking is divided into two categories: 2D Tracking: Described above... (most times referred to only as "tracking", almost all compositing applications support it) 3D Tracking: 3D Tracking attempts to "solve" or derive spatial relationships between points you tracked in 2D space and approximate the distance and parallax between these points in 3D space through the use of complex photogrammetry algorithms. The end result is an approximation of the motion of the camera used to film the scene that can be exported to a 3D application or 2D application to aid in the process of matchmoving. Shots which track most effectively tend to be those with a constant camera path and a significant amount of parallax. Many compositing applications can handle 3D tracking data with varying degrees of proficiency.

/Effects /Post-production

title: Movie Making Manual/Visual Effects


text:
Cookbook:Spirulina

| Basic foodstuffs

Spirulina is a type of cyanobacteria (also known as blue-green algae). It can be purchased at health food stores in tablet, flake, or powder forms.

Spirulina recipes

 * Banana-Spirulina Pancakes

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cookbook:Spirulina


text:
Cookbook:Anise

| Herbs and Spices

Anise is an aromatic plant used as a vegetable and a spice. It should not be confused with star anise, which has a similar flavor profile but is used differently.

Characteristics
Anise seeds are small and brown. The characteristic sweet aroma and flavor comes from the compound anethole, which is also responsible for similar flavors in star anise, fennel and liquorice. Anethole is not soluble in water, so the flavor needs to be released in fat or alcohol in order to flavor a dish well.

The leaves are milder than the seeds and taste similar to tarragon.

Selection and storage
When selecting the seeds, note that they will have a short length of stem remaining. However, avoid seeds that have excessive stems. Store them away from air in a dark, cool place to keep them fresh and prevent loss of flavor.

Uses
The entire anise plant may be used in cooking, including leaves, stems, and seeds. The stalks may be used in stir-fry, and the finely-divided leaves can also be used as an herb. The seeds are often dried and/or ground for long-term storage. A few seeds may be added to pizza sauce to deepen the flavor. Anise seeds can also be processed to make anise essential oil and extract.

Anise is used to flavor a variety of alcoholic beverages, including absinthe, arrak, ouzo, anisette, and sambuca.

title: Cookbook:Anise


text:
Bash Shell Scripting

Wikibooks Contributors Present: Bash Shell Scripting An introductory look to Bash

Currently this book provides an introductory level knowledge of Bash. Go to External Programs, External links and Using man, info and help for further directions and inspirations.

Introduction

 * About Bash
 * History
 * Prerequisites
 * How to read the book?
 * A few notes on terminology
 * Some notes on your shell

Beginning Bash

 * Intro Exercise
 * Simple Commands
 * Exit Status
 * Variables
 * Input-Output
 * Positional Parameters
 * Conditional Expressions
 * Loops
 * Shell Functions
 * Environment

More to Bash

 * Pipelines and Substitution
 * Shell Arithmetic
 * External Programs
 * Whiptail

 * Index of Symbols

Related Wikibooks

 * A Quick Introduction to Unix often involves the Bash shell.
 * Guide to Unix often involves the Bash shell.
 * The Android/Terminal IDE uses Bash as its default command shell.
 * Mac OS X Tiger/A Quick Look Under the Hood uses Bash as its default command shell.
 * Shell Programming
 * Bourne Shell Scripting - Bash is almost Bourne-Shell compatible.

title: Bash Shell Scripting


text:
User:RichardW~enwikibooks

This is a test
First attempt at creating a page.

title: User:RichardW~enwikibooks


text:
Blender 3D: Noob to Pro/Beginning Modeling Final Project

Now that you've gotten the hang of 3D modeling, it's important to get some community feedback on your progress. Don't be an idiot and skip this part, or you'll regret it later. Basically this will help you track your progress and give you something that you'll be working on over a long term and something you'll be proud of.

 * First, you need to come up with a project idea. You can choose your own modeling project, or choose one from the list below.

 * Second, you need to create a model of your idea. Spend a couple of hours on it, and give it some details.

 * Third, once you believe you've come far enough with the model, post it in the Works In Progress forum on BlenderArtists.org(formerly elysiun.com) (you will have to create an account if you haven't already). Post several screenshots of your model from within the Blender (note: creating screenshots is outside the scope of this wikibook, though see note lower down the page).  You can post whatever subject and message with your posting that you would like, or you can use this suggested subject and message:

 * Subject: Beginning Modeling Final Project -

 * Please assist me with any feedback on my model, keeping in mind that I am an absolute beginner still. I appreciate your help.

 * Wait for feedback. It usually comes very quickly.  If you have any questions about feedback that you are given, don't be afraid to ask your questions in the forum.

 * When you and others that have viewed your work feel that you are ready, save your model in some place you can get back to easily. You will continue working on this project once you've learned some new skills.

 * Move on to the next page.

(BTW. in Windows and/or maybe other OS, to take a screenshot press 'PrtScn' (PrintScreen). It will copy the screen to clipboard for you to paste in your favourite graphics application. This may not work in other OSs but try anyway. You can also create Blender screenshot directly from Blender using menu File>Dump 3DView... or File>Dump Screen... )

(In linux under the KDE I use ksnapshot, check under the graphics tab and see if you have it. If not it should be just a google search away :) gl and happy blendering)

(On Mac OS X, press Command (Apple) + Shift + 3 to do a full screen capture)

 * List of ideas:
 * A Computer and keyboard
 * A fishing rod
 * A train engine
 * A skyscraper
 * A robot
 * A Tank (real or made up)
 * An airplane
 * A truck or car
 * Household appliances
 * A Weapon


text:
User talk:Ashl~enwikibooks

'I'm Feeling Lucky' (Google)

 * Hi, Ashl, nice to meet you on Wikibooks.
 * Thank you for pointing out that I'm feeling lucky is in fact always the first site that Google finds, and is not always a random function, like I first thought it was. (Having used Google for five years and more, I should know better[!]). I suppose clicking it over and over is like banging your head against the brick wall - it's so obvious! :-)
 * I'm sure the new users will appreciate that; if they haven't discovered/explored for themselves, like this book (Windows XP for Beginners) encourages them to do.
 * -EuropracBHIT 06:18, 30 Mar 2005 (UTC).

Thanks for Replying Ashl

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Ashl. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Ashl~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:00, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 04:35, 19 April 2015 (UTC)

title: User talk:Ashl~enwikibooks


text: Template:FHSST Physics/Electricity

title: Template:FHSST Physics/Electricity


text:
FHSST Physics/Electricity

= Electricity =

title: FHSST Physics/Electricity


text:
FHSST Physics/Electricity/Flow of Charge

= Flow of Charge =

While the normal motion of "free" electrons in a conductor is random, with no particular direction or speed, electrons can be influenced to move in a coordinated fashion through a conductive material. This motion of electrons is what we call electricity, or electric current. This is in contrast to static electricity, which is an unmoving accumulation of electric charge. Just like water flowing through the emptiness of a pipe, electrons are able to move between the atoms of a conductor. The conductor may appear to be solid to our eyes, but any material composed of atoms is mostly empty space! The liquid-flow analogy is so fitting that the motion of electrons through a conductor is often referred to as a "flow."

As each electron moves uniformly through a conductor, it pushes on the one ahead of it, such that all the electrons move together as a group. The starting and stopping of electron flow through the length of a conductor is virtually instantaneous from one end of a conductor to the other, even though the motion of each electron may be very slow. As an analogy consider a tube filled end-to-end with marbles:

The tube is full of marbles, just as a conductor is full of free electrons. If a single marble is suddenly inserted into this full tube on the left-hand side, another marble will immediately try to exit the tube on the right. Even though each marble only traveled a short distance, the transfer of motion through the tube is virtually instantaneous from the left end to the right end, no matter how long the tube is. With electricity, the overall effect from one end of a conductor to the other effectively instantaneous. Each individual electron, though, travels through the conductor at a much slower pace.

If we want electrons to flow in a certain direction to a certain place, we must provide the proper path for them to move, just as a plumber must install piping to get water to flow where he or she wants it to flow. Wires made of highly conductive metals such as copper or aluminum are used to form this path.

This means that there can be electric current only where there exists a continuous path of conductive material (wire) providing a path for electrons. In the marble analogy, marbles can flow into the left-hand side of the tube (and, consequently, through the tube) only if the tube is open on the right-hand side for marbles to flow out. If the tube is blocked on the right-hand side, the marbles will just "pile up" inside the tube, and marble "flow" will not occur. The same holds true for electric current: the continuous flow of electrons requires there be an unbroken path. Let's look at a diagram to illustrate how this works:

A thin, solid line (as shown above) is the conventional symbol for a continuous piece of wire. Since the wire is made of a conductive material, such as copper, its constituent atoms have many free electrons which can easily move through the wire. However, there will never be a continuous or uniform flow of electrons within this wire unless they have a place to come from and a place to go. Let's add an hypothetical electron "Source" and "Destination:"

Now, with the Electron Source pushing new electrons into the wire on the left-hand side, electron flow through the wire can occur (as indicated by the arrows pointing from left to right). However, the flow will be interrupted if the conductive path formed by the wire is broken:

Air is an insulator, and an air gap separates the two pieces of wire, the path has now been broken, and electrons cannot flow from Source to Destination. This is like cutting a water pipe in two and capping off the broken ends of the pipe: water can't flow if there's no exit out of the pipe.

If we were to take another piece of wire leading to the Destination and simply make physical contact with the wire leading to the Source, we would once again have a continuous path for electrons to flow. The two dots in the diagram indicate physical (metal-to-metal) contact between the wire pieces:

Now, we have continuity from the Source, to the newly-made connection, down, to the right, and up to the Destination. Please take note that the broken segment of wire on the right hand side has no electrons flowing through it, because it is no longer part of a complete path from Source to Destination.

It is interesting to note that no "wear" occurs within wires due to this electric current, unlike water-carrying pipes which are eventually corroded and worn by prolonged flows. Electrons do encounter some degree of friction as they move and this friction can generate heat in a conductor. This is a topic we'll discuss later.


text:
FHSST Physics/Electricity/Circuits

= Circuits =

In order for the Source-and-Destination scheme to work, both would have to have a huge reservoir of electrons in order to sustain a continuous flow! Using the marble-and-tube analogy, the marble source and marble destination buckets would have to be large reservoirs to contain enough marble capacity for a "flow" of marbles to be sustained.

The answer to this paradox is found in the concept of a circuit: a never-ending looped pathway for electrons. If we take a wire, or many wires joined end-to-end, and loop it around so that it forms a continuous pathway ( a circle), we have the means to support a uniform flow of electrons without having to resort to huge reservoirs.

Each electron advancing clockwise in this circuit pushes on the one in front of it, which pushes on the one in front of it, and so on, and so on, just like a hula-hoop filled with marbles. Now, we have the capability of supporting a continuous flow of electrons indefinitely without the need for reservoirs. All we need to maintain this flow is a continuous means of motivation for those electrons, which we'll address in the next section of this chapter.

It must be realized that continuity is just as important in a circuit as it is in a straight piece of wire. Just as in the example with the straight piece of wire between the electron Source and Destination, any break in this circuit will prevent electrons from flowing through it

An important principle to realize here is that it doesn't matter where the break occurs. Any discontinuity in the circuit will prevent electron flow throughout the entire circuit.

A combination of batteries and conductors with other components is called an electric circuit or circuit. The word circuit implies that you return to your starting point and this is an important property of electric circuits. They must contain a closed loop before charge can flow.

The simplest possible circuit is a battery with a single conductor. Now how do we form a closed loop with these two components? The battery has two terminals (connection points). One is called the positive terminal and one the negative. When we describe charge flowing we consider charges moving from the positive terminal of the battery around the conductor and back into the battery at the negative terminal. As much charge flows out of the positive terminal as flows into the negative terminal so there is no build up of charge in the battery. The battery does work on the charges causing them to move round the circuit.

Now I keep talking about batteries and circuits but we need to draw these things to help keep the ideas clear in our minds. To do this we need to agree on how to draw things so that other people can understand what we are doing. We need a convention.

title: FHSST Physics/Electricity/Circuits


text:
FHSST Physics/Electricity/Voltage and Current

= Voltage and current =

As was previously mentioned, we need more than just a continuous path (circuit) before a continuous flow of electrons will occur: we also need some means to push these electrons around the circuit. Just like marbles in a tube or water in a pipe, it takes some kind of influencing force to initiate flow. With electrons, this force is the same force at work in static electricity: the force produced by an imbalance of electric charge.

The electric charge difference serves to store a certain amount of energy. This energy is not unlike the energy stored in a high reservoir of water that has been pumped from a lower-level pond:

The influence of gravity on the water in the reservoir creates a force that attempts to move the water down to the lower level again. If a suitable pipe is run from the reservoir back to the pond, water will flow under the influence of gravity down from the reservoir, through the pipe:

It takes energy to pump that water from the low-level pond to the high-level reservoir, and the movement of water through the piping back down to its original level constitutes a releasing of energy stored from previous pumping.

If the water is pumped to an even higher level, it will take even more energy to do so, thus more energy will be stored, and more energy released if the water is allowed to flow through a pipe back down again:

Electrons are not much different. If we "pump" electrons away from their normal "levels," creating a condition where a force exists as the electrons seek to re-establish their former positions. The force attracting electrons back to their original positions is analogous to the force gravity exerts on water in the reservoir, trying to draw it down to its former level.

Just as the pumping of water to a higher level results in energy being stored, "pumping" electrons to create an electric charge imbalance results in a certain amount of energy being stored in that imbalance. And, just as providing a way for water to flow back down from the heights of the reservoir results in a release of that stored energy, providing a way for electrons to flow back to their original "levels" results in a release of stored energy.

When the electrons are poised in that static condition (just like water sitting still, high in a reservoir), the energy stored there is called potential energy, because it has the possibility (potential) of release that has not been fully realized yet.

This potential energy, stored in the form of an electric charge imbalance and capable of provoking electrons to flow through a conductor, can be expressed as a term called voltage, which technically is a measure of potential energy per unit charge of electrons, or something a physicist would call specific potential energy. Defined in the context of static electricity, voltage is the measure of work required to move a unit charge from one location to another, against the force which tries to keep electric charges balanced. In the context of electrical power sources, voltage is the amount of potential energy available (work to be done) per unit charge, to move electrons through a conductor.

Because voltage is an expression of potential energy, representing the possibility or potential for energy release as the electrons move from one "level" to another, it is always referenced between two points. Consider the water reservoir analogy:

Because of the difference in the height of the drop, there's potential for much more energy to be released from the reservoir through the piping to location 2 than to location 1. The principle can be intuitively understood in dropping a rock: which results in a more violent impact, a rock dropped from a height of one foot, or the same rock dropped from a height of one kilometre? Obviously, the drop of greater height results in greater energy released (a more violent impact). We cannot assess the amount of stored energy in a water reservoir simply by measuring the volume of water any more than we can predict the severity of a falling rock's impact simply from knowing the weight of the rock: in both cases we must also consider how far these masses will drop from their initial height. The amount of energy released by allowing a mass to drop is relative to the distance between its starting and ending points. Likewise, the potential energy available for moving electrons from one point to another is relative to those two points. Therefore, voltage is always expressed as a quantity between two points. Interestingly enough, the analogy of a mass potentially "dropping" from one height to another is such an apt model that voltage between two points is sometimes called a voltage drop.

Voltage can be generated by means other than rubbing certain types of materials against each other. Chemical reactions, radiant energy, and the influence of magnetism on conductors are a few ways in which voltage may be produced. Respective examples of these three sources of voltage are batteries, solar cells, and generators (such as the "alternator" unit under the hood of your automobile). For now, we won't go into detail as to how each of these voltage sources works - more important is that we understand how voltage sources can be applied to create electron flow in a circuit.

Let's take the symbol for a chemical battery and build a circuit step by step:

Any source of voltage, including batteries, have two points for electrical contact. In this case, we have point 1 and point 2 in the above diagram. The horizontal lines of varying length indicate that this is a battery, and they further indicate the direction which this battery's voltage will try to push electrons through a circuit. The fact that the horizontal lines in the battery symbol appear separated (and thus unable to serve as a path for electrons to move) is no cause for concern: in real life, those horizontal lines represent metallic plates immersed in a liquid or semi-solid material that not only conducts electrons, but also generates the voltage to push them along by interacting with the plates.

Notice the little "+" and "-" signs to the immediate left of the battery symbol. The negative (-) end of the battery is always the end with the shortest dash, and the positive (+) end of the battery is always the end with the longest dash. Since we have decided to call electrons "negatively" charged (thanks, Ben!), the negative end of a battery is that end which tries to push electrons out of it. Likewise, the positive end is that end which tries to attract electrons.

With the "+" and "-" ends of the battery not connected to anything, there will be voltage between those two points, but there will be no flow of electrons through the battery, because there is no continuous path for the electrons to move.

The same principle holds true for the water reservoir and pump analogy: without a return pipe back to the pond, stored energy in the reservoir cannot be released in the form of water flow. Once the reservoir is completely filled up, no flow can occur, no matter how much pressure the pump may generate. There needs to be a complete path (circuit) for water to flow from the pond, to the reservoir, and back to the pond in order for continuous flow to occur.

We can provide such a path for the battery by connecting a piece of wire from one end of the battery to the other. Forming a circuit with a loop of wire, we will initiate a continuous flow of electrons in a clockwise direction:

So long as the battery continues to produce voltage and the continuity of the electrical path isn't broken, electrons will continue to flow in the circuit. Following the metaphor of water moving through a pipe, this continuous, uniform flow of electrons through the circuit is called a current. So long as the voltage source keeps "pushing" in the same direction, the electron flow will continue to move in the same direction in the circuit. This single-direction flow of electrons is called a Direct Current, or DC.

Because electric current is composed of individual electrons flowing in unison through a conductor by moving along and pushing on the electrons ahead, just like marbles through a tube or water through a pipe, the amount of flow throughout a single circuit will be the same at any point. If we were to monitor a cross-section of the wire in a single circuit, counting the electrons flowing by, we would notice the exact same quantity per unit of time as in any other part of the circuit, regardless of conductor length or conductor diameter.

If we break the circuit's continuity at any point, the electric current will cease in the entire loop, and the full voltage produced by the battery will be manifested across the break, between the wire ends that used to be connected:

Notice the "+" and "-" signs drawn at the ends of the break in the circuit, and how they correspond to the "+" and "-" signs next to the battery's terminals. These markers indicate the direction that the voltage attempts to push electron flow. Remember that voltage is always relative between two points. Whether a point in a circuit gets labeled with a "+" or a "-" depends on the other point to which it is referenced. Take a look at the following circuit, where each corner of the loop is marked with a number for reference:

With the circuit's continuity broken between points 2 and 3, voltage dropped between points 2 and 3 is "-" for point 2 and "+" for point 3.

Now let's see what happens if we connect points 2 and 3 back together again, but place a break in the circuit between points 3 and 4:

With the break between 3 and 4, the voltage drop between those two points is "+" for 4 and "-" for 3. Take special note of the fact that point 3's "sign" is opposite of that in the first example, where the break was between points 2 and 3 (where point 3 was labeled "+"). It is impossible for us to say that point 3 in this circuit will always be either "+" or "-", because the sign is not specific to a single point, but is always relative between two points!


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FHSST Physics/Electricity/Resistance

= Resistance =

The circuit in the previous section is not a very practical one. In fact, it can be quite dangerous to build (directly connecting the poles of a voltage source together with a single piece of wire). The reason it is dangerous is because the magnitude of electric current may be very large in such a short circuit, and the release of energy very dramatic (usually in the form of heat). Usually, electric circuits are constructed in such a way as to make practical use of that released energy, in as safe a manner as possible.

One practical and popular use of electric current is for the operation of electric lighting. The simplest form of electric lamp is a tiny metal "filament" inside of a clear glass bulb, which glows white-hot ("incandesces") with heat energy when sufficient electric current passes through it. Like the battery, it has two conductive connection points, one for electrons to enter and the other for electrons to exit.

As the electrons work their way through the thin metal filament of the lamp, they encounter more opposition to motion than they typically would in a thick piece of wire. This opposition to electric current depends on the type of material, its cross-sectional area, and its temperature. It is technically known as resistance. (It can be said that conductors have low resistance and insulators have very high resistance.) This resistance serves to limit the amount of current through the circuit with a given amount of voltage supplied by the battery, as compared with the "short circuit" where we had nothing but a wire joining one end of the voltage source (battery) to the other.

When electrons move against the opposition of resistance, "friction" is generated. Just like mechanical friction, the friction produced by electrons flowing against a resistance manifests itself in the form of heat. The concentrated resistance of a lamp's filament results in a relatively large amount of heat energy dissipated at that filament. This heat energy is enough to cause the filament to glow white-hot, producing light, whereas the wires connecting the lamp to the battery (which have much lower resistance) hardly even get warm while conducting the same amount of current.

As in the case of the short circuit, if the continuity of the circuit is broken at any point, electron flow stops throughout the entire circuit. With a lamp in place, this means that it will stop glowing.

As before, with no flow of electrons, the entire potential (voltage) of the battery is available across the break, waiting for the opportunity of a connection to bridge across that break and permit electron flow again. This condition is known as an open circuit, where a break in the continuity of the circuit prevents current throughout. All it takes is a single break in continuity to "open" a circuit. Once any breaks have been connected once again and the continuity of the circuit re-established, it is known as a closed circuit.

What we see here is the basis for switching lamps on and off by remote switches. Because any break in a circuit's continuity results in current stopping throughout the entire circuit, we can use a device designed to intentionally break that continuity (called a switch), mounted at any convenient location that we can run wires to, to control the flow of electrons in the circuit.

This is how a switch mounted on the wall of a house can control a lamp that is mounted down a long hallway, or even in another room, far away from the switch. The switch itself is constructed of a pair of conductive contacts (usually made of some kind of metal) forced together by a mechanical lever actuator or pushbutton. When the contacts touch each other, electrons are able to flow from one to the other and the circuit's continuity is established when the contacts are separated, electron flow from one to the other is prevented by the insulation of the air between, and the circuit's continuity is broken.

In keeping with the "open" and "closed" terminology of circuits, a switch that is making contact from one connection terminal to the other provides continuity for electrons to flow through, and is called a closed switch. Conversely, a switch that is breaking continuity won't allow electrons to pass through and is called an open switch. This terminology is often confusing to the new student of electronics, because the words "open" and "closed" are commonly understood in the context of a door, where "open" is equated with free passage and "closed" with blockage. With electrical switches, these terms have opposite meaning: "open" means no flow while "closed" means free passage of electrons.

title: FHSST Physics/Electricity/Resistance


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FHSST Physics/Electricity/Voltage and Current in a Practical Circuit

= Voltage and current in a practical circuit =

Because it takes energy to force electrons to flow against the opposition of a resistance, there will be voltage manifested (or "dropped") between any points in a circuit with resistance between them. It is important to note that although the amount of current (the quantity of electrons moving past a given point every second) is uniform in a simple circuit, the amount of voltage (potential energy per unit charge) between different sets of points in a single circuit may vary considerably:

Take this circuit as an example. If we label four points in this circuit with the numbers 1, 2, 3, and 4, we will find that the amount of current conducted through the wire between points 1 and 2 is exactly the same as the amount of current conducted through the lamp (between points 2 and 3). This same quantity of current passes through the wire between points 3 and 4, and through the battery (between points 1 and 4).

However, we will find the voltage appearing between any two of these points to be directly proportional to the resistance within the conductive path between those two points, given that the amount of current along any part of the circuit's path is the same (which, for this simple circuit, it is). In a normal lamp circuit, the resistance of a lamp will be much greater than the resistance of the connecting wires, so we should expect to see a substantial amount of voltage between points 2 and 3, with very little between points 1 and 2, or between 3 and 4. The voltage between points 1 and 4, of course, will be the full amount of "force" offered by the battery, which will be only slightly greater than the voltage across the lamp (between points 2 and 3).

This, again, is analogous to the water reservoir system:

Between points 2 and 3, where the falling water is releasing energy at the water-wheel, there is a difference of pressure between the two points, reflecting the opposition to the flow of water through the water-wheel. From point 1 to point 2, or from point 3 to point 4, where water is flowing freely through reservoirs with little opposition, there is little or no difference of pressure (no potential energy). However, the rate of water flow in this continuous system is the same everywhere (assuming the water levels in both pond and reservoir are unchanging): through the pump, through the water-wheel, and through all the pipes. So it is with simple electric circuits: the rate of electron flow is the same at every point in the circuit, although voltages may differ between different sets of points.

When Benjamin Franklin made his conjecture regarding the direction of charge flow (from the smooth wax to the rough wool), he set a precedent for electrical notation that exists to this day, despite the fact that we know electrons are the constituent units of charge, and that they are displaced from the wool to the wax - not from the wax to the wool - when those two substances are rubbed together. This is why electrons are said to have a negative charge: because Franklin assumed electric charge moved in the opposite direction that it actually does, and so objects he called "negative" (representing a deficiency of charge) actually have a surplus of electrons.

By the time the true direction of electron flow was discovered, the nomenclature of "positive" and "negative" had already been so well established in the scientific community that no effort was made to change it, although calling electrons "positive" would make more sense in referring to "excess" charge. You see, the terms "positive" and "negative" are human inventions, and as such have no absolute meaning beyond our own conventions of language and scientific description. Franklin could have just as easily referred to a surplus of charge as "black" and a deficiency as "white," in which case scientists would speak of electrons having a "white" charge (assuming the same incorrect conjecture of charge position between wax and wool).

However, because we tend to associate the word "positive" with "surplus" and "negative" with "deficiency," the standard label for electron charge does seem backward. Because of this, many engineers decided to retain the old concept of electricity with "positive" referring to a surplus of charge, and label charge flow (current) accordingly. This became known as conventional flow notation:

Others chose to designate charge flow according to the actual motion of electrons in a circuit. This form of symbology became known as electron flow notation:

In conventional flow notation, we show the motion of charge according to the (technically incorrect) labels of + and -. This way the labels make sense, but the direction of charge flow is incorrect. Does it matter, really, how we designate charge flow in a circuit? Not really, so long as we're consistent in the use of our symbols. You may follow an imagined direction of current (conventional flow) or the actual (electron flow) with equal success insofar as circuit analysis is concerned. Concepts of voltage, current, resistance, continuity, and even mathematical treatments such as Ohm's Law (chapter 2) and Kirchhoff's Laws (chapter 6) remain just as valid with either style of notation.


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FHSST Physics/Electricity/How Voltage, Current, and Resistance Relate

= How voltage, current, and resistance relate =

First lets recap some of the ideas we have learned so far. We will need these to understand how voltage, current and resistance relate.

An electric circuit is formed when a conductive path is created to allow free electrons to continuously move. This continuous movement of free electrons through the conductors of a circuit is called a current, and it is often referred to in terms of "flow," just like the flow of a liquid through a hollow pipe.

The force motivating electrons to "flow" in a circuit is called voltage. Voltage is a specific measure of potential energy that is always relative between two points. When we speak of a certain amount of voltage being present in a circuit, we are referring to the measurement of how much potential energy exists to move electrons from one particular point in that circuit to another particular point. Without reference to two particular points, the term "voltage" has no meaning.

Free electrons tend to move through conductors with some degree of friction, or opposition to motion. This opposition to motion is more properly called resistance. The amount of current in a circuit depends on the amount of voltage available to motivate the electrons, and also the amount of resistance in the circuit to oppose electron flow. Just like voltage, resistance is a quantity relative between two points. For this reason, the quantities of voltage and resistance are often stated as being "between" or "across" two points in a circuit.

To be able to make meaningful statements about these quantities in circuits, we need to be able to describe their quantities in the same way that we might quantify mass, temperature, volume, length, or any other kind of physical quantity. For mass we might use the units of "pound" or "gram." For temperature we might use degrees Fahrenheit or degrees Celsius. Here are the standard units of measurement for electrical current, voltage, and resistance:

The "symbol" given for each quantity is the standard alphabetical letter used to represent that quantity in an algebraic equation. Standardized letters like these are common in the disciplines of physics and engineering, and are internationally recognized. The "unit abbreviation" for each quantity represents the alphabetical symbol used as a shorthand notation for its particular unit of measurement. And, yes, that strange-looking "horseshoe" symbol is the capital Greek letter Ω, just a character in a foreign alphabet (apologies to any Greek readers here).

Each unit of measurement is named after a famous experimenter in electricity: The ampere after the Frenchman Andre M. Ampere, the volt after the Italian Alessandro Volta, and the ohm after the German Georg Simon Ohm.

The mathematical symbol for each quantity is meaningful as well. The R for resistance and the V for voltage are both self-explanatory, whereas I for current seems a bit weird. The I is thought to have been meant to represent "Intensity" (of electron flow), and the other symbol for voltage, E, stands for "Electromotive force." From what research I've been able to do, there seems to be some dispute over the meaning of I. The symbols E and V are interchangeable for the most part, although some texts reserve E to represent voltage across a source (such as a battery or generator) and V to represent voltage across anything else.

One foundational unit of electrical measurement, often taught in the beginnings of electronics courses but used infrequently afterwards, is the unit of the coulomb, which is a measure of electric charge proportional to the number of electrons in an imbalanced state. One coulomb of charge is equal to 6,250,000,000,000,000,000 electrons. The symbol for electric charge quantity is the capital letter Q, with the unit of coulombs abbreviated by the capital letter "C." It so happens that the unit for electron flow, the ampere, is equal to 1 coulomb of electrons passing by a given point in a circuit in 1 second of time. Cast in these terms, current is the rate of electric charge motion through a conductor.

As stated before, voltage is the measure of potential energy per unit charge available to motivate electrons from one point to another. Before we can precisely define what a "volt" is, we must understand how to measure this quantity we call "potential energy." The general metric unit for energy of any kind is the joule, equal to the amount of work performed by a force of 1 newton exerted through a motion of 1 meter (in the same direction). Defined in these scientific terms, 1 volt is equal to 1 joule of electric potential energy per (divided by) 1 coulomb of charge. Thus, a 9 volt battery releases 9 joules of energy for every coulomb of electrons moved through a circuit.

These units and symbols for electrical quantities will become very important to know as we begin to explore the relationships between them in circuits. The first, and perhaps most important, relationship between current, voltage, and resistance is called Ohm's Law, discovered by Georg Simon Ohm and published in his 1827 paper, The Galvanic Circuit Investigated Mathematically. Ohm's principal discovery was that the amount of electric current through a metal conductor in a circuit is directly proportional to the voltage impressed across it, for any given temperature. Ohm expressed his discovery in the form of a simple equation, describing how voltage, current, and resistance interrelate:

In this algebraic expression, voltage (E) is equal to current (I) multiplied by resistance (R). Using algebra techniques, we can manipulate this equation into two variations, solving for I and for R, respectively:

Let's see how these equations might work to help us analyze simple circuits:

In the above circuit, there is only one source of voltage (the battery, on the left) and only one source of resistance to current (the lamp, on the right). This makes it very easy to apply Ohm's Law. If we know the values of any two of the three quantities (voltage, current, and resistance) in this circuit, we can use Ohm's Law to determine the third.

In this first example, we will calculate the amount of current (I) in a circuit, given values of voltage (E) and resistance (R):

What is the amount of current (I) in this circuit?

In this second example, we will calculate the amount of resistance (R) in a circuit, given values of voltage (E) and current (I):

What is the amount of resistance (R) offered by the lamp?

In the last example, we will calculate the amount of voltage supplied by a battery, given values of current (I) and resistance (R):

What is the amount of voltage provided by the battery?

Ohm's Law is a very simple and useful tool for analyzing electric circuits. It is used so often in the study of electricity and electronics that it needs to be committed to memory by the serious student. For those who are not yet comfortable with algebra, there's a trick to remembering how to solve for any one quantity, given the other two. First, arrange the letters E, I, and R in a triangle like this:

If you know E and I, and wish to determine R, just eliminate R from the picture and see what's left:

If you know E and R, and wish to determine I, eliminate I and see what's left:

Lastly, if you know I and R, and wish to determine E, eliminate E and see what's left:

Eventually, you'll have to be familiar with algebra to seriously study electricity and electronics, but this tip can make your first calculations a little easier to remember. If you are comfortable with algebra, all you need to do is commit E = IR to memory and derive the other two formulae from that when you need them!


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FHSST Physics/Electricity/Ohm's Law Analogy

= An analogy for Ohm's Law =

Ohm's Law also make intuitive sense if you apply it to the water-and-pipe analogy. If we have a water pump that exerts pressure (voltage) to push water around a "circuit" (current) through a restriction (resistance), we can model how the three variables interrelate. If the resistance to water flow stays the same and the pump pressure increases, the flow rate must also increase.

If the pressure stays the same and the resistance increases (making it more difficult for the water to flow), then the flow rate must decrease.

If the flow rate were to stay the same while the resistance to flow decreased, the required pressure from the pump would necessarily decrease.

As odd as it may seem, the actual mathematical relationship between pressure, flow, and resistance is actually more complex for fluids like water than it is for electrons. If you pursue further studies in physics, you will discover this for yourself. Thankfully for the electronics student, the mathematics of Ohm's Law is very straightforward and simple.


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FHSST Physics/Electricity/Power in Electric Circuits

= Power in electric circuits =

In addition to voltage and current, there is another measure of free electron activity in a circuit: power. First, we need to understand just what power is before we analyze it in any circuits.

Power is a measure of how much work can be performed in a given amount of time. Work is generally defined in terms of the lifting of a weight against the pull of gravity. The heavier the weight and/or the higher it is lifted, the more work has been done. Power is a measure of how rapidly a standard amount of work is done.

In electric circuits, power is a function of both voltage and current.

$$P=I E$$

In this case, however, power (P) is exactly equal to current (I) multiplied by voltage (E), rather than merely being proportional to IE. When using this formula, the unit of measurement for power is the watt, abbreviated with the letter "W."

It must be understood that neither voltage nor current by themselves constitute power. Rather, power is the combination of both voltage and current in a circuit. Remember that voltage is the specific work (or potential energy) per unit charge, while current is the rate at which electric charges move through a conductor. Voltage (specific work) is analogous to the work done in lifting a weight against the pull of gravity. Current (rate) is analogous to the speed at which that weight is lifted. Together as a product (multiplication), voltage (work) and current (rate) constitute power.

A circuit with high voltage and low current may be dissipating the same amount of power as a circuit with low voltage and high current. Neither the amount of voltage alone nor the amount of current alone indicates the amount of power in an electric circuit.

In an open circuit, where voltage is present between the terminals of the source and there is zero current, there is zero power dissipated, no matter how great that voltage may be. Since $$P=I E$$ and $$I=0$$ and anything multiplied by zero is zero, the power dissipated in any open circuit must be zero.

Heat as a by-product
Usually heat is produced as a "side-effect" of using an electrical appliance such as a TV set or a computer. Heat is one of the forms of power and there must be a way to get rid of that heat or else the temperature may keep increasing, possibly leading to a breakdown of the apparatus concerned, or even, in an extreme case, to a fire. Make sure there is always enough circulation of air, for examples, in a TV set or a computer; never block the special openings provided in the case of the TV set, or of the computer, etc.; these openings are there on purpose to get rid of the heat.

Often the "rating" of an electrical or electronic item depends on its maximum permitted temperature. A resistor is rated at one watt, for example. Then it will get hot, but not too hot, if its voltage times its current equal no more than one Watt; but that is only true if enough air can get to it to cool it down. Some people will, for example, never "load" a resistor that is rated at a given amount of Watts with more than half that rated power.

The name plate at the rear of the TV set or computer, for example, usually indicates the total amount of Watts that is used, including the part that is converted into heat.


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FHSST Physics/Electricity/Calculating Electric Power

= Calculating electric power =

We've seen the formula for determining the power in an electric circuit: by multiplying the voltage in "volts" by the current in "amps" we arrive at an answer in "watts." Let's apply this to a circuit example:

In the above circuit, we know we have a battery voltage of 18 volts and a lamp resistance of 3 $$\Omega$$. Using Ohm's Law to determine current, we get:

$$I = \frac{E}{R} = \frac {18V}{3\Omega} = 6 A$$

Now that we know the current, we can take that value and multiply it by the voltage to determine power:

$$P = I \ E = (6A) (18V) = 108W $$

Answer: the lamp is dissipating (releasing) 108 watts of power, most likely in the form of both light and heat.

Let's try taking that same circuit and increasing the battery voltage to see what happens. Intuition should tell us that the circuit current will increase as the voltage increases and the lamp resistance stays the same. Likewise, the power will increase as well:

Now, the battery voltage is 36 volts instead of 18 volts. The lamp is still providing 3 $$\Omega$$ of electrical resistance to the flow of electrons. The current is now:

$$I = \frac{E}{R} = \frac {36V}{3\Omega} = 12 A$$

This stands to reason: if $$I = \frac{E}{R}$$, and we double E while R stays the same, the current should double. Indeed, it has: we now have 12 amps of current instead of 6. Now, what about power?

$$P = I \ E = (12A) (36V) = 432W $$

Notice that the power has increased just as we might have suspected, but it increased quite a bit more than the current. Why is this? Because power is a function of voltage multiplied by current, and both voltage and current doubled from their previous values, the power will increase by a factor of 2 x 2, or 4. You can check this by dividing 432 watts by 108 watts and seeing that the ratio between them is indeed 4.

Using algebra again to manipulate the formulae, we can take our original power formula and modify it for applications where we don't know both voltage and resistance:

If we only know voltage (E) and resistance (R):

If we only know current (I) and resistance (R):


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FHSST Physics/Electricity/Resistors

= Resistors =

Because the relationship between voltage, current, and resistance in any circuit is so regular, we can reliably control any variable in a circuit simply by controlling the other two. Perhaps the easiest variable in any circuit to control is its resistance. This can be done by changing the material, size, and shape of its conductive components (remember how the thin metal filament of a lamp created more electrical resistance than a thick wire?).

Special components called resistors are made for the express purpose of creating a precise quantity of resistance for insertion into a circuit. They are typically constructed of metal wire or carbon, and engineered to maintain a stable resistance value over a wide range of environmental conditions. Unlike lamps, they do not produce light, but they do produce heat as electric power is dissipated by them in a working circuit. Typically, though, the purpose of a resistor is not to produce usable heat, but simply to provide a precise quantity of electrical resistance.

The most common schematic symbol for a resistor is a zigzag line:

Resistor values in ohms are usually shown as an adjacent number, and if several resistors are present in a circuit, they will be labeled with a unique identifier number such as $$R_1$$, $$R_2$$, $$R_3$$, etc. As you can see, resistor symbols can be shown either horizontally or vertically:

In keeping more with their physical appearance, an alternative schematic symbol for a resistor looks like a small, rectangular box:

Resistors can also be shown to have varying rather than fixed resistances. This might be for the purpose of describing an actual physical device designed for the purpose of providing an adjustable resistance, or it could be to show some component that just happens to have an unstable resistance:

In fact, any time you see a component symbol drawn with a diagonal arrow through it, that component has a variable rather than a fixed value. This symbol "modifier" (the diagonal arrow) is standard electronic symbol convention.

Because resistors dissipate heat energy as the electric currents through them overcome the "friction" of their resistance, resistors are also rated in terms of how much heat energy they can dissipate without overheating and sustaining damage. Naturally, this power rating is specified in the physical unit of "watts." Most resistors found in small electronic devices such as portable radios are rated at 1/4 (0.25) watt or less. The power rating of any resistor is roughly proportional to its physical size. Also note how resistances (in ohms) have nothing to do with size!

title: FHSST Physics/Electricity/Resistors


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FHSST Physics/Electricity/Nonlinear Conduction

= Nonlinear conduction =

 "Advances are made by answering questions. Discoveries are made by questioning answers."

Bernhard Haisch, Astrophysicist

Ohm's Law is a simple and powerful mathematical tool for helping us analyze electric circuits, but it has limitations, and we must understand these limitations in order to properly apply it to real circuits. For most conductors, resistance is a rather stable property, largely unaffected by voltage or current. For this reason, we can regard the resistance of most circuit components as a constant, with voltage and current being inversely related to each other.

For instance, our previous circuit example with the 3 $$\Omega$$ lamp, we calculated current through the circuit by dividing voltage by resistance (I = E/R). With an 18 volt battery, our circuit current was 6 amperes. Doubling the battery voltage to 36 volts resulted in a doubled current of 12 amperes. All of this makes sense, of course, so long as the lamp continues to provide exactly the same amount of friction (resistance) to the flow of electrons through it: 3 Ω.

However, reality is not always this simple. One of the phenomena we need to mention is of conductor resistance changing with temperature. In an incandescent lamp (the kind employing the principle of electric current heating a thin filament of wire to the point that it glows white-hot), the resistance of the filament wire will increase dramatically as it warms from room temperature to operating temperature. If we were to increase the supply voltage in a real lamp circuit, the resulting increase in current would cause the filament to increase temperature, which would in turn increase its resistance, thus preventing further increases in current without further increases in battery voltage. Consequently, voltage and current do not follow the simple equation $$I=\frac{E}{R}$$ (with R assumed to be equal to 3 $$\Omega$$) because an incandescent lamp's filament resistance does not remain stable for different currents.

The phenomenon of resistance changing with variations in temperature is one shared by almost all metals, of which most wires are made. For most applications, these changes in resistance are small enough to be ignored. In the application of metal lamp filaments, the change happens to be quite large.

A more realistic analysis of a lamp circuit, however, over several different values of battery voltage would generate a plot of this shape:

The plot is no longer a straight line. It rises sharply on the left, as voltage increases from zero to a low level. As it progresses to the right we see the line flattening out, the circuit requiring greater and greater increases in voltage to achieve equal increases in current.

If we try to apply Ohm's Law to find the resistance of this lamp circuit with the voltage and current values plotted above, we arrive at several different values. We could say that the resistance here is nonlinear, increasing with increasing current and voltage. The nonlinearity is caused by the effects of high temperature on the metal wire of the lamp filament.


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FHSST Physics/Electricity/Circuit Wiring

= Circuit wiring =

So far, we've been analyzing single-battery, single-resistor circuits with no regard for the connecting wires between the components, so long as a complete circuit is formed. Does the wire length or circuit "shape" matter to our calculations? Let's look at a couple of circuit configurations and find out:

When we draw wires connecting points in a circuit, we usually assume those wires have negligible resistance. As such, they contribute no appreciable effect to the overall resistance of the circuit, and so the only resistance we have to contend with is the resistance in the components. In the above circuits, the only resistance comes from the 5 $$\Omega$$ resistors, so that is all we will consider in our calculations. In real life, metal wires actually do have resistance (and so do power sources!), but those resistances are generally so much smaller than the resistance present in the other circuit components that they can be safely ignored.

If connecting wire resistance is very little or none, we can regard the connected points in a circuit as being electrically common. That is, points 1 and 2 in the above circuits may be physically joined close together or far apart, and it doesn't matter for any voltage or resistance measurements relative to those points. The same goes for points 3 and 4. It is as if the ends of the resistor were attached directly across the terminals of the battery, so far as our Ohm's Law calculations and voltage measurements are concerned. This is useful to know, because it means you can re-draw a circuit diagram or re-wire a circuit, shortening or lengthening the wires as desired without appreciably impacting the circuit's function. All that matters is that the components attach to each other in the same sequence.

It also means that voltage measurements between sets of "electrically common" points will be the same. That is, the voltage between points 1 and 4 (directly across the battery) will be the same as the voltage between points 2 and 3 (directly across the resistor). Take a close look at the following circuit, and try to determine which points are common to each other:

Here, we only have 2 components excluding the wires: the battery and the resistor. Though the connecting wires take a convoluted path in forming a complete circuit, there are several electrically common points in the electrons' path. Points 1, 2, and 3 are all common to each other, because they're directly connected together by wire. The same goes for points 4, 5, and 6.

The voltage between points 1 and 6 is 10 volts, coming straight from the battery. However, since points 5 and 4 are common to 6, and points 2 and 3 common to 1, that same 10 volts also exists between these other pairs of points:

Between points 1 and 4 = 10 volts

Between points 2 and 4 = 10 volts

Between points 3 and 4 = 10 volts (directly across the resistor)

Between points 1 and 5 = 10 volts

Between points 2 and 5 = 10 volts

Between points 3 and 5 = 10 volts

Between points 1 and 6 = 10 volts (directly across the battery)

Between points 2 and 6 = 10 volts

<tt>Between points 3 and 6 = 10 volts </tt>

Since electrically common points are connected together by (zero resistance) wire, there is no significant voltage drop between them regardless of the amount of current conducted from one to the next through that connecting wire. Thus, if we were to read voltages between common points, we should show (practically) zero:

<tt>Between points 1 and 2 = 0 volts        Points 1, 2, and 3 are </tt>

<tt>Between points 2 and 3 = 0 volts        electrically common </tt>

<tt>Between points 1 and 3 = 0 volts </tt>

<tt>Between points 4 and 5 = 0 volts        Points 4, 5, and 6 are </tt>

<tt>Between points 5 and 6 = 0 volts        electrically common </tt>

<tt>Between points 4 and 6 = 0 volts </tt>

This makes sense mathematically, too. With a 10 volt battery and a 5 $$\Omega$$ resistor, the circuit current will be 2 amperes. With wire resistance being zero, the voltage drop across any continuous stretch of wire can be determined through Ohm's Law as such:

It should be obvious that the calculated voltage drop across any uninterrupted length of wire in a circuit where wire is assumed to have zero resistance will always be zero, no matter what the magnitude of current, since zero multiplied by anything equals zero.

Because common points in a circuit will exhibit the same relative voltage and resistance measurements, wires connecting common points are often labeled with the same designation. This is not to say that the terminal connection points are labeled the same, just the connecting wires. Take this circuit as an example:

Points 1, 2, and 3 are all common to each other, so the wire connecting point 1 to 2 is labeled the same (wire 2) as the wire connecting point 2 to 3 (wire 2). In a real circuit, the wire stretching from point 1 to 2 may not even be the same color or size as the wire connecting point 2 to 3, but they should bear the exact same label. The same goes for the wires connecting points 6, 5, and 4.

Knowing that electrically common points have zero voltage drop between them is a valuable troubleshooting principle. If I measure for voltage between points in a circuit that are supposed to be common to each other, I should read zero. If, however, I read substantial voltage between those two points, then I know with certainty that they cannot be directly connected together. If those points are supposed to be electrically common but they register otherwise, then I know that there is an "open failure" between those points.

One final note: for most practical purposes, wire conductors can be assumed to possess zero resistance from end to end. In reality, however, there will always be some small amount of resistance encountered along the length of a wire, unless it's a superconducting wire. Knowing this, we need to bear in mind that the principles learned here about electrically common points are all valid to a large degree, but not to an absolute degree. That is, the rule that electrically common points are guaranteed to have zero voltage between them is more accurately stated as such: electrically common points will have very little voltage dropped between them. That small, virtually unavoidable trace of resistance found in any piece of connecting wire is bound to create a small voltage across the length of it as current is conducted through. So long as you understand that these rules are based upon ideal conditions, you won't be perplexed when you come across some condition appearing to be an exception to the rule.


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FHSST Physics/Electricity/Polarity of Voltage Drops

= Polarity of voltage drops =

We can trace the direction that electrons will flow in the same circuit by starting at the negative (-) terminal and following through to the positive (+) terminal of the battery, the only source of voltage in the circuit. From this we can see that the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again.

As the current encounters the 5 $$\Omega$$ resistance, voltage is dropped across the resistor's ends. The signs of this voltage drop is negative (-) at point 4 with respect to positive (+) at point 3. We can mark the polarity of the resistor's voltage drop with these negative and positive symbols, in accordance with the direction of current (whichever end of the resistor the current is entering is negative with respect to the end of the resistor it is exiting:

We could make our table of voltages a little more complete by marking the polarity of the voltage for each pair of points in this circuit:

Between points 1 (+) and 4 (-) = 10 volts

Between points 2 (+) and 4 (-) = 10 volts

Between points 3 (+) and 4 (-) = 10 volts

Between points 1 (+) and 5 (-) = 10 volts

Between points 2 (+) and 5 (-) = 10 volts

Between points 3 (+) and 5 (-) = 10 volts

Between points 1 (+) and 6 (-) = 10 volts

Between points 2 (+) and 6 (-) = 10 volts

<tt>Between points 3 (+) and 6 (-) = 10 volts </tt>

While it might seem a little silly to document polarity of voltage drop in this circuit, it is an important concept to master. It will be critically important in the analysis of more complex circuits involving multiple resistors and/or batteries.

It should be understood that polarity has nothing to do with Ohm's Law: there will never be negative voltages, currents, or resistance entered into any Ohm's Law equations! There are other mathematical principles of electricity that do take polarity into account through the use of signs (+ or -), but not Ohm's Law.


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FHSST Physics/Electricity/Series and Parallel

= What are "series" and "parallel" circuits? =

Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.

There are two basic ways in which to connect more than two circuit components: series and parallel. First, an example of a series circuit:

Here, we have three resistors (labeled $$R_1$$, $$R_2$$, and $$R_3$$), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling - those little numbers to the lower-right of the letter "R" - are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The defining characteristic of a series circuit is that there is only one path for electrons to flow. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4.

Now, let's look at the other type of circuit, a parallel configuration:

Again, we have three resistors, but this time they form more than one continuous path for electrons to flow. There's one path from 8 to 7 to 2 to 1 and back to 8 again. There's another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there's a third path from 8 to 7 to 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through $$R_1$$, $$R_2$$, and $$R_3$$) is called a branch.

The defining characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points.

And, of course, the complexity doesn't stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too:

In this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through $$R_1$$ (from point 2 to point 1). In this configuration, we'd say that $$R_2$$ and $$R_3$$ are in parallel with each other, while $$R_1$$ is in series with the parallel combination of $$R_2$$ and $$R_3$$.

This is just a preview of things to come. Don't worry! We'll explore all these circuit configurations in detail, one at a time!

The basic idea of a "series" connection is that components are connected end-to-end in a line to form a single path for electrons to flow:

The basic idea of a "parallel" connection, on the other hand, is that all components are connected across each other's leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to flow, but only one voltage across all components:

Series and parallel resistor configurations have very different electrical properties. We'll explore the properties of each configuration in the sections to come.


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FHSST Physics/Electricity/Simple Series Circuits

= Simple series circuits =

Let's start with a series circuit consisting of three resistors and a single battery:

The first principle to understand about series circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path for electrons to flow in a series circuit, and because free electrons flow through conductors like marbles in a tube, the rate of flow (marble speed) at any point in the circuit (tube) at any specific point in time must be equal.

From the way that the 9 volt battery is arranged, we can tell that the electrons in this circuit will flow in a counter-clockwise direction, from point 4 to 3 to 2 to 1 and back to 4. However, we have one source of voltage and three resistances. How do we use Ohm's Law here?

An important caveat to Ohm's Law is that all quantities (voltage, current, resistance, and power) must relate to each other in terms of the same two points in a circuit. For instance, with a single-battery, single-resistor circuit, we could easily calculate any quantity because they all applied to the same two points in the circuit:

Since points 1 and 2 are connected together with wire of negligible resistance, as are points 3 and 4, we can say that point 1 is electrically common to point 2, and that point 3 is electrically common to point 4. Since we know we have 9 volts of electromotive force between points 1 and 4 (directly across the battery), and since point 2 is common to point 1 and point 3 common to point 4, we must also have 9 volts between points 2 and 3 (directly across the resistor). Therefore, we can apply Ohm's Law ($$I =\frac{E}{R}$$) to the current through the resistor, because we know the voltage (E) across the resistor and the resistance (R) of that resistor. All terms (E, I, R) apply to the same two points in the circuit, to that same resistor, so we can use the Ohm's Law formula with no reservation.

However, in circuits containing more than one resistor, we must be careful in how we apply Ohm's Law. In the three-resistor example circuit below, we know that we have 9 volts between points 1 and 4, which is the amount of electromotive force trying to push electrons through the series combination of $$R_1$$, $$R_2$$, and $$R_3$$. However, we cannot take the value of 9 volts and divide it by 3k, 10k or 5k $$\Omega$$ to try to find a current value, because we don't know how much voltage is across any one of those resistors, individually.

The figure of 9 volts is a total quantity for the whole circuit, whereas the figures of 3k, 10k, and 5k $$\Omega$$ are individual quantities for individual resistors. If we were to plug a figure for total voltage into an Ohm's Law equation with a figure for individual resistance, the result would not relate accurately to any quantity in the real circuit.

For $$R_1$$, Ohm's Law will relate the amount of voltage across $$R_1$$ with the current through $$R_1$$, given $$R_1$$'s resistance, 3k$$\Omega$$:

But, since we don't know the voltage across $$R_1$$ (only the total voltage supplied by the battery across the three-resistor series combination) and we don't know the current through $$R_1$$, we can't do any calculations with either formula. The same goes for $$R_2$$ and $$R_3$$: we can apply the Ohm's Law equations if and only if all terms are representative of their respective quantities between the same two points in the circuit.

So what can we do? We know the voltage of the source (9 volts) applied across the series combination of $$R_1$$, $$R_2$$, and $$R_3$$, and we know the resistances of each resistor, but since those quantities aren't in the same context, we can't use Ohm's Law to determine the circuit current. If only we knew what the total resistance was for the circuit: then we could calculate total current with our figure for total voltage ($$I =\frac{E}{R}$$).

This brings us to the second principle of series circuits: the total resistance of any series circuit is equal to the sum of the individual resistances. This should make intuitive sense: the more resistors in series that the electrons must flow through, the more difficult it will be for those electrons to flow. In the example problem, we had a 3 k$$\Omega$$, 10 k$$\Omega$$, and 5 k$$\Omega$$ resistor in series, giving us a total resistance of 18 k$$\Omega$$:

In essence, we've calculated the equivalent resistance of $$R_1$$, $$R_2$$, and $$R_3$$ combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of $$R_1$$, $$R_2$$, and $$R_3$$:

Now we have all the necessary information to calculate circuit current, because we have the voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 k$$\Omega$$):

Knowing that current is equal through all components of a series circuit (and we just determined the current through the battery), we can go back to our original circuit schematic and note the current through each component:

Now that we know the amount of current through each resistor, we can use Ohm's Law to determine the voltage drop across each one (applying Ohm's Law in its proper context):

Notice the voltage drops across each resistor, and how the sum of the voltage drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts. This is the third principle of series circuits: that the supply voltage is equal to the sum of the individual voltage drops.

However, the method we just used to analyze this simple series circuit can be streamlined for better understanding. By using a table to list all voltages, currents, and resistances in the circuit, it becomes very easy to see which of those quantities can be properly related in any Ohm's Law equation:

The rule with such a table is to apply Ohm's Law only to the values within each vertical column. For instance, $$E_R1$$ only with $$I_R1$$ and $$R_1$$ $$E_R2$$ only with $$I_R2$$ and $$R_2$$  etc. You begin your analysis by filling in those elements of the table that are given to you from the beginning:

As you can see from the arrangement of the data, we can't apply the 9 volts of $$E_T$$ (total voltage) to any of the resistances ($$R_1$$, $$R_2$$, or $$R_3$$) in any Ohm's Law formula because they're in different columns. The 9 volts of battery voltage is not applied directly across $$R_1$$, $$R_2$$, or $$R_3$$. However, we can use our "rules" of series circuits to fill in blank spots on a horizontal row. In this case, we can use the series rule of resistances to determine a total resistance from the sum of individual resistances:

Now, with a value for total resistance inserted into the rightmost ("Total") column, we can apply Ohm's Law of $$I=\frac{E}{R}$$ to total voltage and total resistance to arrive at a total current of 500 A:

Then, knowing that the current is shared equally by all components of a series circuit (another "rule" of series circuits), we can fill in the currents for each resistor from the current figure just calculated:

Finally, we can use Ohm's Law to determine the voltage drop across each resistor, one column at a time:


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FHSST Physics/Electricity/Simple Parallel Circuits

= Simple parallel circuits =

Let's start with a parallel circuit consisting of three resistors and a single battery:

The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across $$R_1$$ is equal to the voltage across $$R_2$$ which is equal to the voltage across $$R_3$$ which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values:

Just as in the case of series circuits, the same caveat for Ohm's Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm's Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:

At this point we still don't know what the total current or total resistance for this parallel circuit is, so we can't apply Ohm's Law to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor ("branch") currents:

As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through $$R_1$$, some more splits off at point 6 to go up through $$R_2$$, and the remainder goes up through $$R_3$$. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through $$R_1$$, $$R_2$$, and $$R_3$$ join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through $$R_1$$, $$R_2$$, and $$R_3$$.

This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the $$I_T$$ spot on our table with the sum of $$I_{R1}$$, $$I_{R2}$$, and $$I_{R3}$$:

Finally, applying Ohm's Law to the rightmost ("Total") column, we can calculate the total circuit resistance:

Please note something very important here. The total circuit resistance is only 625 $$\Omega$$: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:

The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.


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FHSST Physics/Electricity/Power Calculations

= Power calculations =

When calculating the power dissipation of resistive components, use any one of the three power equations to derive and answer from values of voltage, current, and/or resistance pertaining to each component:

This is easily managed by adding another row to our familiar table of voltages, currents, and resistances:

Power for any particular table column can be found by the appropriate Ohm's Law equation (appropriate based on what figures are present for E, I, and R in that column).

An interesting rule for total power versus individual power is that it is additive for any configuration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure of rate of work, and since power dissipated must equal the total power applied by the source(s) (as per the Law of Conservation of Energy in physics), circuit configuration has no effect on the mathematics.