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IB Physics/Astrophysics SL

F.1.1 Structure of the Solar System
Within our own universe, there exist many celestial bodies each with their own unique properties; the planets being no exception.

Note: All planets beyond Mars are gas giants; i.e. Jupiter is a failed sun.

With these given properties and the right equation, you can find the volume and density of any of these planets.

Volume:$$V=\frac{4}{3}\pi r^3$$ r=radius of the planet or sphere

Density:$$D= \frac{M}{V}$$ M=mass V=volume

F.1.2 Bodies within the Universe
Binary Star: Two stars orbiting a common center.

Black Dwarf: The remnant of a white dwarf after it has cooled down. It has very low luminosity.

Black Hole: A singularity in space-time: the end result in the evolution of a very massive star.

Brown Dwarf: Gas and dust that did not reach high enough temperatures to initiate fusion. These objects continue to compact and cool down.

Cepheid Variable: A star of variable luminosity. The Luminosity increases sharply and falls off gently with a well-defined period. The period is related to the absolute luminosity of the star and so can be used to estimate the distance to the star by using the Cepheid variable as a standard candle.

Clusters of Galaxies: Two or more Galaxies that are close enough to each other that they affect each others through gravitation.

Comet: A small body composed of mainly ice and dust that orbits the sun in an elliptical orbit.

Constellation: A group of stars which are in a particular pattern or design.

Clusters: Gravitationally bound system of galaxies/stars.

Constellations: Group of galaxies/stars given a specific name. The 12 zodiacs are examples - Pisces, Aries, Taurus, Gemini, Cancer, and so on.

Dark Matter: Matter in galaxies that is too cold to radiate. Its existence is inferred from techniques rather then direct visual contact.

Galaxies: Giant assemblies of stars, gas, and dust held together by the gravitational forces they have on each other. Our Galaxy is called the Milky Way.

Interstellar Medium: Gases and dust that are filling the space between stars. Interstellar mass’s density is very low with about one atom of gas for every cubic centimeter of space.

Main Sequence Star: A normal star that is undergoing nuclear fusion of hydrogen into helium.

Neutron stars: A very dense star, consisting only of uncharged neutrons. They are created when very massive stars explode, leaving this neutron 'ball' behind. A neutron star is smaller then a white dwarf and extremely dense. It is microscopic and is a prime example of microscopic quantum physics.

Nebulae: From the Latin word for 'cloud'. Used to label all sorts of stuff in space, that are now known as star cluster or galaxies. It is sometimes still used for a concentration of gas and dust.

Nova: A sudden increase in luminosity of a white dwarf caused by material from a nearby star falling into the white dwarf.

Parallax: The apparent motion of a star against the background of a more distant star, due to the motion of the Earth around the Sun. The angle is measured at different times during the year. The distance of the Sun to the Earth is known. Distances are specified in parallax angles in seconds of arc (parsec). At large distances the uncertainty becomes too large and it can't be applied. Example : angle = (6_10-5)° = (6_10-5)° (3600) = 0.22´´ of arc (seconds of arc) in parsecs: 1/0.22´´ = 4.5 pc.

Planetary nebula: The ejected envelope of a red giant star.

Pulsar: Sends out sharp, strong burst of radio waves at regular intervals ranging between milliseconds and 4 seconds. They appear to be rapidly rotating highly magnetic neutron stars. The pulses are very energetically charged particles. The rotation and pulse rates gradually slow down as energy is radiated away.

Quasars: Small, extraordinarily luminous extragalactic objects with high redshifts. They do not seem to conform to Hubble’s Law. They are as bright as nearby stars, but display very large redshifts. According to Hubble’s law the quasars must be either extremely distant and incredibly bright (thousands of times brighter than ordinary galaxies) or that they are closer than the redshift suggests. There is either an unresolved brightness problem or an unresolved redshift problem. One theory is that quasars could be powered by black holes.

Red Dwarf: A very small star with low temperature. It is relatively red in color.

Red Giant: Luminous stars with low surface temperature. These stars are produced when hydrogen in the core of the star has fused into more heavy helium. Gravity forces the star to contract, but at the same time it heats up. The hydrogen around the core now burns more fiercely and causes the outer envelope of the star to expand and thus cool. This low surface temperature produces light at a longer wavelength.

Stellar Cluster: A group of stars that are physically near each other in space, created by a collapse of the same gas cloud.

Supernova: Gigantic stellar explosions. These occur when very massive stars explode. Because of Einstein's law, E=mc2, fusion can not take place after that iron is created. The star collapses since there is no force that can outweigh the gravitational force. When it is impossible to compress it further, it explodes violently giving out extremely luminous light.

White Dwarfs: Are developed as all fuel in a star is exhausted. Gravity forces the star (with a mass lower than 1.4 solar masses) to contract and heat up at a high pressure. The atoms lose some of their electrons. It becomes a small hot white dwarf.

F.1.3 Stellar Cluster and Constellations
A stellar cluster is a group of stars that are near each other in space and created from the same gas cloud. On the other hand, a constellation is just a group of stars that seem to be close to each other because they form a recognizable pattern in space. In a stellar cluster the stars are attracted to each other because of high temperatures in the core of the stars, where the protons end up having electrostatic repulsion due to the Hydrogen Nuclei Fusion. A constellation appears to be stars close to each other. The celestial sphere is what makes them appear like the stars are physically near.[LG,JD]

F.1.4 Light Years
A light year is a unit of measurement of Ultra-solar system distances. It's the distance traveled by light in one year. The speed of light is 186,287.5 miles per second. You can find out the number of seconds in a year by multiplying the number of seconds in a minute (60) by the number of minutes in an hour (60). Then multiply that by the number of hours in a day (24), and multiply that by the number of days in a year (approximately 365.25). One light year is equal to 9.46 x 10^15m, which is also equal to 0.3068 Parsecs(Pc).

Example: The distance to the nearest star (Proxima Centauri)from the Earth is 4.31 light years, which is equal to 1.3pc. This means that it would take 4.3 years to send a message to Proxima Centauri.

F.1.5 Relative Distances
The average distance between stars in a galaxy is approximately 1 pc, which is equal to 3.26 light years. The average distance between galaxies within the same cluster ranges from 100 kpc (kilo-parsecs) to several hundred kpc (kilo-parsecs). Galaxies in different clusters can be up to a few Mpc (mega-parsecs) apart. 1 Mpc is equal to 1000 kpc.

F.1.6 Movement of Constellations and Stars over time
During the night, stars and constellations seem to be moving from east to west, but the relative position of the constellations do not change. The celestial sphere is the area around earth where the stars and constellations are located. The North Star, Polaris, is on the north celestial pole and doesn't seem to move at all. The rest of the stars and constellations seem to rotate around the North Star. Stars seem to move less the further north or south they are located. As the Earth rotates, the hemispheres receive different views of the stars and constellations.

For the Earth to make one revolution per year there must be a small change every day. The change per night is 0.986 degrees and is not easily detected.

F.2.1 Energy Source
All stars follow a simple proton-proton cycle in order to maintain an equilibrium between gravity and pressure. When the star is expending fuel it rises in temperature and therefore rises in pressure. This is required in order to keep a balance between the great force of gravity that is trying to compress the star. At the beginning of a star's life cycle the star consists mainly of hydrogen; in fact they are made of 98% hydrogen. There are three basic stages of the Proton-Proton cycle: 1.) Two hydrogens fuse to form a deuterium, plus a positron and a neutrino. Each positron is annihilated to create 2γ(gamma) particles [they are in turn absorbed and re-emitted as 200,000 photons of light per γ(gamma) particle.] 2.a.) A deuterium and a hydrogen fuse to create helium and a γ(gamma) particle. 2.b.) Another deuterium and a hydrogen fuse to create helium and a γ(gamma) particle. [Thus far there have been 4γ(gamma) particles created, those are 800,000 photons of light. That's one bright star!] 3.) Two helium atoms fuse to create a heavy helium atom. Once the hydrogen in the star runs out, it begins to consume the created helium from the hydrogen reactions. Based on the star's color, you can find out what type of fuel it's consuming. This is where the Hertzsprung-Russell diagrams were derived from.

F.2.2 Star stability and Equilibrium
The stability of a star depends on the equilibrium between two opposing forces. The equilibrium depends on the gravitation which can collapse the star and the radiation pressure which can make the star expand. This equilibrium is gained through nuclear fusion which provides the energy the star needs to keep it hot so that the star's radiation pressure is high enough to oppose gravitational contraction.

F.2.3 Luminosity of a star
"Luminosity is the amount of energy radiated by the star per second; that is, it is the power radiated by the star. As shown in the next section, luminosity depends on the surface temperature and surface area of the star." In simple words: Luminosity depends on temperature and the radius (surface area) of the star.

Luminosity is the result of a chemical reaction within a star.

F.2.4 Brightness and Measurement of Luminosity
Brightness is the received energy per second per unit area by apparent brightness.

b=L/4πd-2 Wm-2

Apparent brightness is directly proportional to intrinsic luminosity, where apparent brightness is measured using a charge coupled device (CCD) that releases one electron when hit by a photon.

F.2.5 Black Body Radiation
A black body is an object that absorbs all electromagnetic radiation that falls onto it. The Stefan-Boltzmann Law can be used to determine the radiation being emitted in the form of electromagnetic waves by the black body. To understand the Stefan-Boltzmann Law in further depth refer to section F.2.8. The Wien displacement law relates the wavelength to surface temperature. The Wein constant is 2.90 x 10^-3 Km. This means that the higher the temperature, the lower the wavelength at which most of the radiation is emitted as energy.

F.2.6 Spectra of black bodies
As the overall intensity increases, the relative intensity of the peak of the curve (most intense wavelength) decreases in intensity, and the most intense wavelength gets longer and longer (less and less energetic).

F.2.7 Wien's Law
Wein's displacement law states that the higher the temperature the lower the wavelength at which most of the energy is radiated.

Wein's displacement law is stated as: λmax= b/T Where λ is the maximum wavelength (meters), T is the temperature of the blackbody (Kelvin), and b is a constant for 2.90 x 10-3Km

F.2.8 Stefan-Boltzmann Law
The Stefan-Boltzmann Law helps to find luminosity of a star's distance from an observer. The luminosity is then used to find the absolute brightness of the star. The equation to find luminosity is L=σAT4.

The Stefan-Boltzmann constant: σ = 5.67 x 10-8Wm-2K-4

A= Surface Area

T= Temperature

Once the luminosity is found, place it into the equation b=L/4πd2 for L.

d= distance for this equation.

This will allow you to find the absolute brightness of the star.

F.2.9 Stellar Spectra
The surface temperature of a star is determined by measuring the wavelength at which most of the radiation is emitted. Most stars have essentially the same chemical composition., yet show different absorption spectra. The reason is that different stars have different temperatures. Absorption spectra gives information about the temperature of the star and its chemical composition. Stars are divided into seven spectral classes according to their color.

Spectral Class Summary

Spectral Class Effective Temperature (K) Colour H Balmer Features Other Features M/MSun R/RSun L/LSun Main Sequence Lifespan O 28,000 - 60,000 Blue weak ionised He+ lines, strong UV continuum 20 - 60 9 - 15 90,000 - 800,000 1 - 10 Myr

B 10,000 - 28,000 Blue-white medium neutral He lines 3 - 18 3.0 - 8.4 95 - 52,000 11 - 400 Myr

A 7,500 - 10,000 White strong strong H lines, ionised metal lines 2.0 - 3.0 1.7 - 2.7 8 -55 400 Myr - 3 Gyr

F 6,000 - 7,500 White-yellow medium weak ionised Ca+ 1.1 - 1.6 1.2 - 1.6 2.0 - 6.5 3 - 7 Gyr

G 4,900 - 6,000 Yellow weak ionised Ca+, metal lines 0.85 - 1.1 0.85 - 1.1 0.66 - 1.5 7 - 15 Gyr

K 3,500 - 4,900 Orange very weak Ca+, Fe, strong molecules, CH, CN 0.65 - 0.85 0.65 - 0.85 0.10 - 0.42 17 Gyr

M 2,000 - 3,500 Red very weak molecular lines, eg TiO, neutral metals 0.08 - 0.05 0.17 - 0.63 0.001 - 0.08 56 Gyr

It is known that hydrogen is the predominant element in normal main sequence stars, making up 70% of their mass followed by helium with 20%. The rest is made up of heavier elements.

F.2.13 Hertzsprung-Russell diagram
Determines the fate of stars.

F.3.1 Parallax
Parallax is a term used to describe the distance between two objects in space. When an observer on Earth photographs a relatively nearby star against a background of distant stars on two different occasions six months apart, the target star image will appear to have shifted against the more distant stellar background. The baseline shift of the observer on Earth is 2 astronomical units (AU). By convention, calculations are normalized to one AU, the radius of the Earth's orbit, so one half of the measured shift in apparent position is deemed the "parallax" of the target.

F.3.2 The Parsec
A parallax of one arcsecond is called a parsec. Since we know, the radius of the Earth's orbit, simple Euclidean geometry allows us to calculate that a star exhibiting a one arcsecond shift is 3.26 light years or one parsec away from Earth.

F.3.3 Limits of the Stellar parallax method
Hubble's law: v = Hd, where v is the speed, H is the Hubble parameter, and d is the distance. It describes Hubble's observation, that most lines in the spectra of other galaxies were redshifted, and the amount of shift was approximately proportional to the distance of the galaxy from us. So the velocity is proportional to the distance. H is approximately 75km/s/Mpc. Note that this does not work well for nearby galaxies because of two reasons. One the error margin for calculating distances is somewhere around 15% and we have more accurate ways to measure distances for close distances and two earth is still a part of many of the closer systems so the difference in velocities is negligible. It's also important to note that this is a value that changes with new data coming in. Hubble's law really describes the speed at which celestial bodies move away from each other and changes because the further away a galaxy gets the faster it goes.

F.3.4 Determining distance with Stellar parallax method
Determining distances with stellar parallax is done by measuring a star periodically against a fixed background. Then you do some simple algebra with the change in area. Much more accurate at close distances than Hubble's law but Hubble's law is much better for long distances despite its error margin.

F.3.5 Apparent Magnitude Scale
The U.S. physicists A. Penzias and R. Wilson detected (in 1956) microwave radiation coming equally from all directions in the sky, day and night. This radiation is like the one radiated by a black body at a temperature of 3 Kelvin, therefore the name 3K radiation.

This discovery supports the theory of Big Bang, where strong shortwave radiation was supposed to be sent out. The radiation spread filling the expanding universe uniformly. With time it cooled, to the now observed temperature of 3K, and now strikes the Earth as microwaves.

F.3.6 Concept of Absolute Magnitude and Determining Absolute Magnitude
The Absolute Magnitude of a star relies on the apparent magnitude initially. The apparent magnitude of a star is a measure of brightness of a star seen from earth in a relative system of classification. The higher the numerical value of apparent magnitude, the dimmer the star. This relates to absolute magnitude because the absolute magnitude is the apparent magnitude a start would have if observed from a distance of 10pc. To determine absolute magnitude, you use the following equation : M=m+2.5log(100/d2)

F.3.7 Apparent Brightness and Apparent Magnitude Examples
An example for apparent magnitude is as follows: The star Capella has an apparent magnitude of +0.05 and its distance from Earth is 14pc. Estimate its absolute magnitude.

To solve this problem we would use the following equation: M=m+2.5log(100/d2)

Next we would plug in the variables and get the following: M=+0.05+2.50log(100/142)

M=2.55log(100/196)= -.745252982

So the estimated absolute magnitude is -.745252982

F.3.8 Finding Luminosity from the Spectrum
Luminosity is the amount of energy radiated by the star per second. Basically put, it is the power radiated by the star. Luminosity depends on the surface temperatue and the surface area of the star. To relate luminosity to the spectrum, the higher the luminosity a star has, the higher up on the spectrum it will be.

F.3.12 Cepheid Variables and Cepheid Nature
Cephied Variable:  A star thats luminosity increases sharply and falls gently in a period of time, the period of time is related to the luminosity of the star. The Cephied variable is used to estimate the distance of the star. There is a relationship between the period of the light curve and the peak luminosity. On Cephieds have periods of one to fifty days. The variable provides information of the star structure developing theories for stellar structures.

F.3.13 Cepheid Variables and relation to Period and Average Absolute Magnitude
In absolute and apparent magnitudes astronomers would classify stars according to how they appeared to be to the observer on earth based on its brightness. The higher the magnitude then the dimmer the star, brightness of stars were defined in assigned numbers that varied the magnitude of the start from 1-6. Not all stars are the same distance, which is a factor in how the magnitude of a star is classified, while the brightness also varies as opposed to two starts in the same place, direction, and same brightness.

F.3.14 Cepheid Variables as "Standard Candles"
"Standard Candles," are those that are described as Cepheid Variables because the way that the Cepheid stars are described, it occurs the same way as candles. The farther away a candle is from the other the less luminosity and brightness is seen from the observer. Distance is what takes away the luminosity and brightness of a star the farther away it is from the observers stand point of view.

F.3.15 Cepheid Variables on the Luminosity-Period Graph
Cepheid Variables on a Luminosity-Period Graph due to their brightness increase and gradual fade offs, curves on the graph. It gives a sine graph type of picture because of it's periodic behavior in brightness. The outer layers of the star go through contractions and expansions periodically. When it expands outward it's because the star is brighter at a surface at high velocity, and when it's dim at a surface it moves inward.

F.4.1 Newton's Theory of the Universe
If the universe is infinite, how come then that we observe a black night sky? Shouldn't such a universe provide an infinite number of stars so that wherever we looked it would be bright?

F.4.2 Olber's Paradox
The Olbers Paradox questions why the night sky is dark. If the universe was infinite and thereby contained an infinite amount of stars, then theoretically there would be an infinite amount of energy radiating from the stars making the night sky infinitely bright. The Olbers Paradox applies to all infinite models, but does not apply to finite models. This is due to the fact that:
 * 1) There is a finite number of stars and each star has a finite lifetime.
 * 2) The universe has a finite age and stars that are beyond the event horizon have not yet had time for their light to reach Earth.
 * 3) The radiation received is redshifted and so contains less energy.

F.4.3 Solutions to Olber's Paradox
The matter and radiation of our present time was initially all packed together into an extremely hot and dense fireball, that exploded giving rise to the Big Bang. Within seconds, matter was accelerated through 3 dimensions, expanding and developing very rapidly. Time became a measure of the rate of that expansion, the necessary 4th dimension.

F.4.4 Doppler Red Shift and Expanding Universe
Curvature of space (also called four-dimensional space-time) : According to Einstein's general theory of relativity light is also affected by gravity. This means that light can be bent 'around' planets, and follow a curved path. Light must travel through the shortest distance available between two points, meaning that the curved path is the shortest distance, hence space itself is curved.

Open universe : The universe will continue to expand forever because the curvature of the universe is negative.

Flat universe : The curvature is zero, the universe is infinite.

Closed universe : The curvature is positive, so the universe is finite.

This can all be showed through non-Euclidean mathematics, where the sum of angles in a triangle either subtends of exceeds 180°.

F.4.5 Big Bang Model
Big Bang model: About ten to twenty billion years ago,all matter and energy in the universe was concentrated in one area from which it expanded quickly. This expansion began when an explosion took place somewhere in space. There is no gravity in space acting on the debris of this explosion so it moves away from the site of the explosion at a velocity that is proportional to its distance from the spot where the explosion took place. Therefore, the farther away the debris is from the explosion, the faster it moves and likewise the closer it is the slower it moves.

The big bang model is related to the four fundamental forces which include: gravity, strong force, weak force, and electromagnetic force. It is assumed that at first these forces started out as one and later separated into the four.

F.4.9 Critical Density
The density that is required to create a flat universe is called critical density. It is round about 5×10^-26kgm^-3.

title: IB Physics/Astrophysics SL


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IB Physics/Optics SL

H.1.1
Specular reflection : This is illustrated by mirrors, where a reflected image is formed as a result. This is caused by very smooth surfaces, which reflect light back evenly across their entire surface.

Diffuse reflection : This is the more common example of light bouncing of anything else to form an image off that object (i.e. when light bounces off a piece of paper, we see the paper not a reflection). This is caused by the fact that the surface of the paper is not smooth, but rather light is reflected in all directions.

If you look at a mirror from different angles and you see different reflections, where as the paper looks the same from all angles. Diffuse reflection allows us to see things, since otherwise we could only see objects emitting light. As a result, we can see anything which isn't pitch black (i.e. absorbs all light) when there's enough light around.

H.1.2
Formation of virtual images in plane mirrors

Light travels out in every direction form every point on the object, but it's impossible to draw them all, so a ray diagram is used. It is usual to have the object as an arrow on a slope, pointing up, though anything which will show both rotation and inversion will be sufficient.

Draw two rays from a point, slightly diverging (separating) which then strike the mirror, and then reflecting (angle of refraction = angle of incidence).

Draw a line perpendicular to the mirror, and the angle of entry one one side equals the angle of exit on the other). These two lines then move away from the mirror and reach the 'eye' (just draw something to represent it, it isn't an art competition).

The two rays reaching the eye are then extended back up 'inside' the mirror to the point where they meet (should be on the same level as the original object). This point is where the image is formed. It should be upright but 'flipped' around a vertical axis, and is a virtual image.

Finally, draw the image inside the mirror.

H.1.3
Mirrors are used in optical instruments, mostly to let us see things which aren't actually in front of us (or viable normally).
 * Bike mirrors allow us to see the car about to run us down.
 * Periscopes allow submarines to see, and then blow up ships on the surface.
 * A mirror on the moon was used to measure it's distance from earth by bouncing a laser off it and measuring the time for it to return.

H.1.4
Radius of curvature : If the curve of a lens was extended into a full circle, the radius of curvature is the radius of that circle. Radius of curvature is only relevant for spherical lenses, but they're the only ones we consider anyway. The focal length of a spherical mirror will be half the radius of curvature, or the radius of curvature will be 2F, depending on how you want to look at it.

Principal axis : The principal axis is a line perpendicular to the curved surface of the lens at its centre (i.e. a line running horizontally through the center of the lens).

Principal focus : This value is related to the focal length of the lens. The principal focus is always one focal length away from the centre of the lens (therefore there are two principal foci, one on either side, usually designated F on the far side, and F' on the same side as the object). Parallel rays of light entering the lens either converge towards the principal focus (convex lenses) or diverge from it (concave lenses).

Focal length : Focal length is the shortest distance between the principal axis and the principal focus.

Paraxial rays : Paraxial rays are those that are close to the principal axis and parallel to it.

Magnification : The magnification is usually defined as image height/object height, representing in effect how much bigger the image is. This is also directly related to image distance/object distance.

H.1.5
Virtual image : Defined as and image one which doesn't actually have any light rays running through it, only 'virtual' rays (i.e. the ones extended inside the mirror).

Real images : Those which do have light rays running through them (those which can be shown on a screen).

As for drawing ray diagrams to help analyze this, I suppose that falls under the 'if light rays go through it' definition. Real images are generally used in projection systems, where the image is being displayed on a screen (or cast directly into the eye). Virtual images are often used, for example, in magnifying lenses, where the image must be visible from different angles.

H.1.6
The mirror equation : 1/d o + 1/d i = 1/f. Based on two of image distance, object distance and focal length, the third can be found. This equation is in the data book under the Optics section.

H.1.7
Thin lenses : Used to converge or diverge light within optical systems. Telescopes, for example, focus the light from stars onto our eyes, at the end of the telescope, and magnifying glasses diverge the light, forming a magnified virtual image on the other side.

Acoustic lenses : focus sound in much the same way an optical lens focuses light. Snell's law describes the refraction of sound as it passes through an interface between two materials of differing sound speed. An acoustic lens provides the appropriate material thicknesses that focus a parallel wavefront of sound to a single focal point.

Microwave lenses : Presumably something to do with microwaves, which are really just another part of the em spectrum, so they will do the exact same thing with lenses as viable light. I suppose in microwave ovens you use diverging lenses to scatter the microwaves around, but someone else may be able to elaborate.

Drawing ray diagrams
 * First draw in the principal axis, and the lens (convex or concave as appropriate) in the centre. Mark points on the principal axis on both sides for the focal point and 2x the focal point (F and 2F).
 * Draw the object (again usually an arrow) on the left.
 * Draw two rays from the top of the arrow. The first runs parallel to the principal axis until it hits the lens, then goes in a straight line down through the focal point (for a convex lens) or goes in a straight line away for the focal point on the object's side (for a concave lens). The second ray runs from the top of the object to the centre of the lens, where it meets the principal axis, and continues straight through.
 * If these two rays meet on the other side, then this is where the image will be formed (i.e. where you should put the screen). If they diverge, then trace both line back from the lens to where they meet on the other side. This forms a virtual image on the same side as the object, which must be viewed without a screen from the opposite side of the lens. If the object was on the focal point, then the two rays will be parallel, and no image will be formed.

Sign conventions : Basically, heights above the principal axis are positive, below are negative. The focal length of a converging lens is considered positive, and diverging NEGATIVE (So if you studied it to be positive, just remember it's NEGATIVE for a DIVERGING lens focal point). The object distance is positive if it is on the same side the light is coming from (almost always the case) and the image distance is positive if it is on the opposite side to where the light is coming from (in general, positive for real images, negative for virtual).

H.1.8
Deriving the lens equation is done using the similar triangles formed by the ray passing through the centre of the lens. Triangle #1 is formed by the object, the principal axis and this ray, and triangle #2 is formed by the image, the principal axis and the ray. This allows us to get the relationship Hi/H o = Di/D o, since all sides of the triangles are in proportion.

The second pair of triangles are formed by the ray going down from the lens through the focal point (as in drawing the diagrams). This forms a triangle #1 with the lens (height is equal to Ho), the axis and the ray. Triangle #2 is the image, the axis and the ray. This allows us to get the relationship Hi/H o = (Di-f)/f. Note, the top term is (f-Di) for a diverging lens, but this is eliminated because f is negative.

These two equations can then be equated and simplified to form 1/d o + 1/d i = 1/f, the lens equation (yes, it's the same as the mirror equation). This is much clearer with diagrams, so perhaps someone would like to add some.

As can be seen also from the first equation, Di and Do are also directly related to the ratio of heights. i.e. the magnification, M=Hi/H o =Di/D o.

H.1.9
Simple magnifier (i.e. a magnifying glass) : This uses a convex lens, where the object is placed closer than F to the lens. This results in a magnified, virtual image behind the lens (actually further than the object, but it's bigger to compensate). This allows the object to appear bigger than it is. This is used, for example, to read words in small print.

Compound microscope : Produces a magnified, upside down image by using two lenses. The first lens, close to the object, is called the objective, while the second is called the eyepiece. The object is placed further than Fobjective away form the objective lens. This then converges the light to form an image closer than Feye piece to the eyepiece lens. this results in the light diverging out of the eyepiece, and so by tracing back, we can find the position of the magnified virtual image.

Astronomical telescope : This is virtually identical to a compound microscope, except this time the light is entering not from a close object, but in parallel lines at an angle to the lens (since it is coming from so far away). The objective lens focuses the light at a point closer than Fe which then diverges it, and forms a magnified, upside down image behind the lens.

H.1.10
Total internal reflection prisms and optical fibres

(See also section 11.1.4)

Total internal reflection (TIR) : Total internal reflection is a result of Snell's law, and occurs when the angle of refraction is greater than 90. Light is traveling in a substance when it meets a boundary with a less dense medium on the other side. The light will be refracted away from the normal. There comes a point, however, where the resulting angle of refraction is 90. SinØc = n2/n1 (since sin 90 = 1, it can be cancelled out). If the angle of incidence is equal to Øc, then the light will travel along the surface of the more dense medium, however if it is greater then all the light is reflected back into the more dense medium.

TIR prisms : When light travelling inside a prism reaches a boundary and attempts to leave, it is subject to total internal reflection if the angle of incidence is greater than Øc. As such, a (right angle triangular) prism can be designed where light enters on the vertical side, is totally internally reflected against the hypotenuse, and then leaves through the bottom side. This can be used to form a sort of periscope without using mirrors, which means 100% of the light is transmitted (mirrors tend to lose some). Also, such a prism could be designed so that, when placed with it's hypotenuse down, light enters one side and is refracted downward, TIRs on the bottom side and then leaves the other side. This means that rays entering the prism higher up, come out further down (because they strike the bottom further along). This can be used, therefore, to invert an image (as in the telescope described above).

Optical fibers : Optical fibers consist basically of a very thin 'wire' of a material with a high refractive index, into which light is sent. Since it has a high index, even when the fiber is bent, the critical angle will still not be exceeded, and all light will be totally internally reflected. This can then be used to transmit light very easily around corners etc. Since the light travels very quickly without resistance, it is preferable to using electric current in wires, and is often used in telecommunications. These fibres can also be used to carry light into the human body, and as in an endoscope, to allow us to examine the inside of the human body without cutting it open ( or doing any other gross biology stuff ). Decorative lamps are a rather trivial use, but you can use them for that to.

Note, in the first two applications, it is important that fibres next to each other are optically insulated, so no light can possibly jump from one to another. This is usually done by coating them in a material of lower refractive index than the fibre.

H.1.11
Thanks to [mailto:wksslask@box43.gnet.pl Arkadiusz Staekiewicz].

Determine the minimum deviation angle for dispersion in a prism and explain how the dispersion effect is used in spectroscopes.

A prism separates white light into a rainbow of colours, because the index of refraction of a material depends on the wavelength. White light is a mixture of all visible wavelengths, and when incident on a prism, the different wavelengths are bent to varying degrees. Because the index of refraction is greater for the shorter wavelength, violet light is bent the most and red the least as indicated. This spreading of white light into the full spectrum is called dispersion.

A spectroscope is a device to measure wavelengths accurately using a diffraction grating or a prism to separate different wavelengths of light. A prism, because of dispersion, bends light of different wavelengths into different angles. A prism has the disadvantage that it produces less sharp lines and is less able to separate closely spaced lines. But it has the advantage of deflecting more light (and so is more useful for dim sources) than a typical diffraction grating, since with the latter most of the light passes through to the central peak. An important use of any of these devices is for the identification of atoms or molecules. When a gas is heated or a large electric current is passed through it, the gas emits a characteristic line spectrum. That is, only certain wavelengths of light are emitted, and these are different for different elements and compounds.

Minimum deviation by a prism occurs when the refraction is symmetrical. The minimum deviation D appears in the equation: n=[sin1/2(A+D)]/[sin1/2A], where A is the principal refracting angle of the prism i.e. the angle made by the refracting faces in a principal section perpendicular to the refracting edge, n is the refractive index. For narrow-angle prisms, D=(n-1)A  (approximately), and over a moderate range of angle of incidence there is an approximately constant deviation.

H.2.1
interference and diffraction in various apparatus.

In each case, monochromatic, coherent light is assumed to be striking the apparatus in question.

Single slit : Light is assumed to strike a thin slit in a medium, and as a result, form an infinite number of point sources along that slit, each radiating light in a semi-circular pattern. We then consider light traveling from these points at a number of critical angles.
 * 1) Light traveling straight through remains in phase, and so produces a bright band on the screen level with the slit.
 * 2) Light traveling out at such an angle that the light from the top source must travel exactly 1 wavelength further than the bottom one to reach the screen. This means that there must be a source in the center which must travel 1/2 of a wavelength further than the bottom one. As a result, this 1/2 wavelength and the top ray are completely out of phase, and so they destructively interfere. The same can be said about the ray immediately below the top, and the one immediately below the middle, and so we continue down until all the rays have destructively interfered, causing a dark band on the screen at this angle ( sinØ = λ/slit length ).
 * 3) Now consider the point where the top ray has 3/2x λ further to go. The bottom and middle thirds of the slit all destructively interfere as in 2, but the top third goes on to the screen, causing another (though smaller) light band ( sinØ = 3 x λ/2 x slit length ).
 * 4) The point where the top ray has 2 x λ further to go. All rays destructively interfere ( sinØ = 2 x λ/slit length ) and so on. This causes a series of bright and dark bands, with the bright bands becoming smaller and smaller as the angle from the center becomes greater.

Double slit : This is effectively the same as above, only simpler as we have two point sources, rather than a row of them. When the difference in distance from the screen to the two slits is an even number of wavelengths, then a bright band occurs, when it has a 1/2λ on the end, then it causes a dark band. In general, both follow the equation D sinØ = m x λ, where D is either the distance between the two slits, or the length of the slit.

Diffraction grating : Again similar to the double silt, only this time we have many slits. This produces a far sharper set of peaks, because with two slits, when the angle is slightly away from a bright band, the two sources are only slightly out of phase, and so will still produce a reasonably bright band. With many slits, a slight change in angle will produce a far greater drop in intensity since one slit can be completely out of phase with another several hundred slits away. This results in the bright bands being very sharp, and makes a diffraction grating far more precise for calculating wavelength than double slits.

Phase change (this definition is needed for next two) : Whenever light changes mediums, some of it will be reflected by the boundary. If the light enters a medium of higher refractive index than the one it was travelling in, then a phase change will occur (i.e. in a transverse wave model, peaks become troughs and vice versa). This sometimes complicates the calculation of when light will destructively interfere, as any phase changes must be taken into account along with the path difference.

Thin films : The classic example of this is a thin layer of oil (assumed to have lower refractive index than water) floating on top of water. This produces a sort of rainbow effect in the right light conditions. When light enters the oil, some of it is reflected (with a phase change). The remaining light continues down and some is reflected of the oil-water boundary (again with a phase change, meaning the two can be ignored in this case. If the film is like a soap bubble, only one phase change will occur, and it must be accounted for). This means that if the film is a certain thickness, certain wavelengths will be reinforced will others will destructively interfere (this is how they make those sun glasses which look red from the outside etc.). Note that the light is always assumed to enter and leave vertically, though it will be easier to draw at an angle, this should be noted with any diagram. It may be necessary to think of the angle involved if the question wants fringes on the film rather than certain wavelengths being reinforced/destructively interfering though.

Newton's rings : In Newton's rings, there is a flat glass surface with a curved glass plate (think of the bottom part of a sphere being cut off) placed on top of it. This means the gap between the two pieces of glass increases going further out from the centre. Light is reflect off the bottom of the curved plate (with no phase change) and off the top of the base plate (with a phase change). This means that to reinforce, the actual difference between the two distance travelled must be (k+1/2) x λ (where k is some whole number). Note that this means that at the very center there will be a dark spot, not a bright spot (as with the various slit ones above).

H.2.2
For single slits, the minima will be defined by m x λ = a sin Ø (where m is a whole number which changes depending on the order of the fringe, and a is the width of the slit). For double slits, or gratings, m x λ = d sin Ø = xd/D (where d is the distance between the center of two consecutive slits, x is the bandwidth (distance between consecutive bright bands on the screen) and D is the distance to the screen).

H.3.1
Light normally travels in all planes (i.e. one wave goes up and down, another goes left and right and at every angle in between). Polarised light is light which only travels in one of these planes (i.e. only up and down). This can occur due to a variety of stuff (discussed in the next section), and can usually be analysed using a polaroid (like the stuff they make sun glasses out of) because when it is rotated, the intensity of light passing through will change.

H.3.2
The different ways of getting polarised light.

By reflection : As was discussed above, when light strikes a boundary between two media, some will be reflected and some will enter the second media. If we think about it though, light moving parallel to the surface is more likely to be reflected than light moving perpendicular (i.e. up and down across the boundary. This occurs completely when the difference between the angle of the reflected light and the refracted light is exactly 90 degrees. You can work it out, but this means that tanØp=n (assuming we're going from a medium with refractive index 1 to one with n) where Øp is the angle at which light must enter (angle of incidence) to be fully polarised.

By double refraction : Before the idea of refraction was even understood, it was known that some crystals (such as Iceland spar) would refract two separate rays of light, and so they were called doubly refracting. When another type of crystal (tourmaline) was rotated, the light from this would either be transmitted or blocked, depending on orientation. It was only much later that this effect was understood as polarisation.

By selective absorption : Certain materials, known as polaroids, are capable of polarising light when it passes through them. They are created by long chains of molecules arranged such that they are all parallel, leaving small slits between. The axis of the polaroid is the direction of the slits, and when unpolarised light passes through, light moving in the plane of the lists is transmitted completely, while light in the perpendicular plane is completely absorbed. Vector diagrams can be used to find the component of light in the appropriate plane, and hence the amount transmitted.

By scattering : As light travels through the atmosphere, the waves will interact with some air molecules (specifically O2 and N2). The electric field associated with the light is absorbed, causing the electric charges within the molecule to oscillate. Oscillating electric charges cause electromagnetic radiation to be produced perpendicular to the direction of oscillation. As a result, an observer looking directly up will see polarised light, where the plane of polarisation is perpendicular to the direction of oscillation (and the direction the light was initially traveling). From other viewing angles, the light will be partially polarised. This can be extended to explain why the sky is blue (because short wavelengths are affected more) and why sunsets are red (because almost all the blue has already been wiped out), but I don't think it's necessary to know that.

H.4.1
Experiments which confirm the wave-particle duality of light:

Interference : This is the double slit thing from above. This is a wave property, as particles can not destructively interfere (or interfere at all really).

Diffraction : Again as above. Particles would not diffract when passing through a thin slit, but waves would.

Polarisation : Particles would not be oscillating, and so could not be polarised, but light can be, so it must be a wave (maybe).

The Photoelectric effect (See also 13.5.1) : When electromagnetic radiation above a certain frequency shines on a metal surface, a current is produced. The wave theory of light predicts that energy is carried within the light in a continuous form (as it is in all waves). This suggests that any frequency of electromagnetic radiation will eventually cause the emission of electrons (and so produce a current) if it is applied to the metal for long enough, however this does not occur. Rather, nothing can make electromagnetic radiation below this frequency cause emission (not increases in intensity, leaving it on for long periods or anything else). Below fo (the 'certain' frequency), an increase in intensity increases the number of electrons given off per second, but the maximum Ek of the electrons is not changed by anything except frequency. All this is measured by using a variable voltage opposing the current created by the photoelectron emission, and using a galvanometer to measure when the current is exactly zero.

All this information suggests that the energy within light is quantized, and thus light is broken into individual particles called photons, which carry a certain amount of energy related to their frequency. Each atom can therefore accept only one photon, not two added together. This also explains why the Ek of the electrons is determined by the wavelength, because the Ek comes from the energy remaining after the electron has been excited out of the atom. Therefore light is a particle, but also a wave (and so on).

Cromption effect : This effect deals with the collision of short wavelength (high frequency) light such as X-rays with electrons. Crompton used the photon model of light to predict that the electron would be scattered by this collision, and the x-rays remaining afterwards would have less energy. This was demonstrated mathematically using the laws of conservation of energy and momentum, and practically by experimental results. The wave theory predicts that no shift in the energy of the resulting x-rays would occur, because the electromagnetic radiation of frequency f would be absorbed by the electrons, causing it to oscillate with frequency f and re-emit the x-rays with the same frequency. Once again, this is evidence for the particle nature of light.

H.5.1
Uses of lasers :

Technology : Bar code scanners (relies on the reflection or absorption of laser light), laser disks (light will either be reflected or scatters by 'pits' on the surface of the disk, allowing data to be read from it.

Industry : Surveying (because the beam is very intense, it can travel over large distances without losing intensity, so can be used to measure distances etc.), welding (intense light is very hot, and also able to be focused very sharply to a point), ditto for machining metals and drilling tiny holes.

Medicine : Burning away tissue in small areas (very hot and precise), breaking up gall + kidney stones (can easily be carried inside an optical fiber, and so can be inserted in the body), attaching the retina, cauterising lymph vessels and capillaries (again, precise, focused heat).

H.5.2
Holography is the making of holograms, which are 3 dimensional images embedded in a 2 dimensional surface. This process is performed using lasers (due to the fact that they are focused, coherent sources, as opposed to normal light). A wide laser beam shines on a half silvered mirror, so half the light goes through to the film, while the other half is reflected down on to the object to go into the hologram. Light from every point on the object, as a result, strikes every point on the film, and the interference of the two beams allow the film to record both the intensity and the phase of the light reaching it. If we think about it, the intensity bit is just like a normal photograph. The phase relates directly to the 'depth' (because the phase changes over distance). The interference between the rays from the object and the rays going straight through the mirror allows this phase difference to be found, and so recorded on the film.

To view the hologram a laser must once again be used. After the film is developed, the hologram is placed in laser light of the same wavelength as is was produced with. The hologram acts as a sort of diffraction grating, producing a real 2D image of the hologram (on the opposite side to the laser), and a virtual 3D image on the same side as the laser, thus producing the 3D effect.

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

title: IB Physics/Optics SL


text:
New Zealand History/New Zealand Government

‹ Back to New Zealand Prime Ministers of the Twenty First Century

The New Zealand Government at the Turn of the Twenty First Century

New Zealand, at the turn of the Twenty First Century, was run by the Labour Party, under the leadership of Helen Clark - the electoral system being MMP (Mixed Member Proportional Representation), which was implemented in 1996.

The MMP system allows for two votes on election day: a 'Party Vote' (the vote given to the party the voter wants represented in Parliament), and the 'Electorate Vote' (the vote given to the MP the voter wants to represent their electorate in Parliament). Due to the nature of this system, it is unlikely that a party will be able to govern without the support of minor parties. When parties make an agreement to form a Government it is called a 'coalition'. For example, following the 2005 election, the formal coalition comprised of the Labour Party (the major party, receiving a majority of votes) and the Progressive Party (the minor party), but the New Zealand First and United Future parties also provided confidence and supply to Labour in return for their leaders being Ministers outside of cabinet. The rest of the parties made up the Opposition: after the 2005 election, this was the National Party, the ACT Party and the Maori Party.

In 2008, the National Party received the majority of votes in the election, and it made a coalition agreement with the ACT Party, Maori Party, and United Future, which collectively made up enough seats in Parliament to have an overall majority. Therefore, a National-led coalition became the new Government. This made the Labour Party, Green Party, and the Progressive Party the Opposition.

''This table shows how Parliament was made up after the 2005 election. It also gives a background to some of the main parties in New Zealand politics, and who they aim to represent in Parliament.''

Forward to the Bibliography ›


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Hindi/Consonant combinations

This page lists frequently occurring conjunct consonants of the devanagari script. The way this page displays depends on your browser's rendering support and the availability of appropriate fonts.

Special cases
Combinaisons des consonnes (hindi)

title: Hindi/Consonant combinations


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User talk:59.101.225.67

--Az1568(Talk) 08:23, 6 November 2007 (UTC)

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User talk:124.106.226.136

--AdRiley

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User talk:124.6.169.46

--AdRiley 12:28, 6 November 2007 (UTC)

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User talk:62.171.194.13

-- Herby talk thyme 12:40, 6 November 2007 (UTC)

Blocked
You were warned and are now blocked from editing, thanks -- Herby talk thyme 12:45, 6 November 2007 (UTC)

title: User talk:62.171.194.13