text:
IB Physics/Wave Phenomena

11.1.1
I assume what we're talking about here is y = Asin(wt ± kx), the equation in the data book. This can be used to describe a traveling wave as follows.

The amplitude is A (because sine curves range from 1 to -1, multiplying by A will make it range from A to -A). w is defined as 2 x π x f and k is defined as 2 x π/λ. The value of t will shift the whole curve to the left or right (assuming the curve is positive in the middle, increasing t moves it to the left, decreasing to the right).

The period of the curve will be defined by the wavelength and frequency in the equation. It's a good idea to play around with this on a graphing calculator to get a feel for it. Anyway, the equation is obviously used for modeling waves.

11.1.2
This is effectively what we already described in the standard level section, but is repeated below for convenience.

Displacement vs time : This graph tracks the movement of a particle as a wave moves through it. With displacement on the vertical axis, and time on the horizontal. The particle will move up and down in a sine curve type pattern. This graph allows us to find both frequency (which will be the number of crests in 1 sec) and period (which will be the time between crests), but tells us nothing about the wave speed or wavelength.

Displacement vs position : This is basically a 'snapshot' of the displacement of all the particles going through the medium at a given time. Displacement is on the vertical axis, and position (or ie distance from an arbitrary origin in the material) is on the x. The distance between peaks represents the wavelength. The wave speed can not be calculated directly from this graph, but can be found by combining the information from this and the displacement vs time graph (as described in the next section).

11.1.3
Huygens' principle is a geometrical representation of how waves move through media. Each wave front is assumed to be an infinite number of point sources, each radiating in a circle. After a given period of time, a new wave front is drawn along the edges of these radiated circles, and the process is repeated.

To draw it on paper, start with a wave front, place a number of points, and from these, draw the waves being emitted as if each of these were a point source. When unobstructed, this results in a series of circles, but obstructions can change things. Waves could be reflected or absorbed by an object, waves entering a medium of higher optical density will slow down (and so won't go as far).

After a given period of time (depends on the speed of the wave), draw an new wave front running along the edges of these circles as appropriate for the situation. The process is repeated over and over until it gets so boring that you stop. This sort of diagram helps to explain some of the phenomena of waves.

 * Diffraction : A very thin slit will only have a single point source, and so it will radiate in a circle, or wraps around an object, but you really need to draw a diagram to see that.

 * Refraction : As the wave enters the more dense medium at an angle, the leading edge slows down, pulling the wave around to a new angle.

This model can be applied to any waves but in questions they will probably be light, water or sound.

11.1.4
Partial reflection occurs when ever light changes media. For example, when light goes from water to air, some light is reflected from the boundary (the same occurs from air to water).

Total internal reflection occurs when light enters a boundary (from the more dense side) at an angle greater than the critical angle, and all the light is refracted back into the original medium. This critical angle can be found by inserting 90 as the angle of refraction in Snell's law, thus creating n1 x sin ic = n2. Any wave striking at an angle of incidence above this critical angle will totally internally reflect. When the angle of incidence equals the critical angle, the light will run along the boundary, and below it, refraction (and partial reflection) will occur as usual.

11.1.5
Light through optical fibres : This is used both as a communication system, and as a sort of camera in hard to reach places. Light is totally internally reflected through the glass core, which can be bent as long as the light passing through it does not exceed the critical angle (see the optics option for more info).

Prismatic reflectors : Glass has a critical angle above 45, and so it is possible to use a glass, right angled, triangular prism as a reflector. Light enters the longest side, bounces off one side, off the other, then out the way it came in...this is more effective than using a mirror because 100% of the light is reflected, where as mirrors are never 100% efficient. This set up can also be rearranged to build a periscope (light goes an and out the two short sides, bouncing off the long one) without mirrors.

Air near hot surfaces : For example, the pavement seems to be wet and shiny on a exceptionally hot day. Since hot air rises, there is a "gradient" of temperatures of air. Since air's refractive index changes with temperature, Light waves travelling through this will refract according to the temperatures of air, causing the light from the sun to gradually defract with a elliptical path - "Bending" From its original (incident) angle, to being nearly horizontal, to finally travelling upwards into your eye. The virtual rays thus create an illusion that the light source (the sun)is on the floor.

11.1.6
Refractive index is dependent on the wavelength of the wave, thus different wavelengths of light will be refracted different amounts through the same boundary. Short wavelength light will be refracted more, and long wave length less. This means that if white light is shone onto a prism, then the light can be separated out into it's component colours, red being refracted the least, and violet the most.

11.2.1
This first bit might seem familiar for the standard level notes.

If, for example, we have two point sources producing waves in a circle, they will interfere differently at different points. The easiest way to do this is to draw circles out from the source representing the crests (Except now we can call this Huygen's principle). When two of these coincide, constructive interference produces a bigger crest. When two gaps coincide, we get a bigger trough, when one crest and one trough coincide, there is destructive interference, and they add to zero. This allows the interference pattern, and the amplitude at each point to be found.

Also relevant to the discussion of Huygen's principle is that fact that these point sources effectively produce a wave front, since other parts of the wave interfere destructively, thus demonstrating how exactly Huygen's principle can be accounted for (beyond being a geometric representation).

11.2.2
For two sources to be coherent, they must emit identical frequency waves, in the same phase (i.e. when one emits a crest, so must the other). Path difference is the difference between the distances of a certain point from each source. If the path difference is a multiple of the wavelength, then constructive interference (an anti-node) is produced, if it's a multiple + λ/2 complete destructive interference occurs (producing a node). Points in between these two have something between a node and an anti-node. The pattern produced is a series of lines pointing away from the point exactly between the sources, and alternating constructive-destructive-constructive out from the centre.

11.2.3
Light strikes the two slits, and then produces two coherent point sources next to each other.

 * Light striking the center of the screen has an equal path difference from both, and so produces a bright band on the screen level with the slit (since the light is spread over the smallest area).

 * Light travelling out at such an angle that the light from the top source must travel exactly λ/2 further than the bottom one to reach the screen. This means the two waves are out of phase, and destructively interfere on the screen (producing a dark spot). As we move further around, the path difference will be 1 wavelength, they will reinforce, and produce a bright band, and so on alternating.

This experiment can be defined by the equation m x λ = d sin Ø = xd/D, where d is the distance between the centres of the two slits, x is the bandwidth (distance between consecutive bright bands on the screen) and D is the distance to the screen. The xd/D bit assumes a curved screen, but it's find for a flat screen so long at you're not too far from the centre. I don't know if this is really necessary, but see the optics option section for more detail.

11.2.4
These notes comet out of the optics section, so there might be too much detail.

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 as a result of the following. 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. If the film is like a soap bubble, however, 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. Note, 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 plate (think of the bottom part of a sphere 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 reflected 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 distances travelled must be k+λ/2, where k is some integer. Note, this means that at the very centre there will be a dark spot, not a bright spot (as with the various slit ones above).

Since the syllabus says no experimental details will be required, most of that probably isn't necessary.

11.2.5
A diffraction grating is basically a series of slits, rather than two (as in Young's double slit). These slits produce much more precise lines, because rather than just requiring two beams to coincide, they require many to do so. This produces a much sharper pattern, and is more easy to analyse. If white light goes through the defraction grating, different frequencies will diffract different amounts, and so spectra will be produced. Using this the component colours of light can be found, with their exact wavelengths (because it affects the angle at which the bright bands occur). Calculations can be done with m x λ = d sin Ø = xd/D, where d is the distance between the centre of two consecutive slits, x is the bandwidth (distance between consecutive bright bands on the screen) and D is the distance to the screen.

Also relevant here is a quick explanation of the diffraction pattern for each single slit (as this 'defines an envelope on the interference patters' i.e. it shows what it will be under). There is a large wide peak of intensity in the centre, dropping to zero, followed by a series of smaller peaks of half the width of the central one. Each minima for this is defined by D sin Ø = m x λ, where D is the width of each slit. I don't know if they really want much detail on this.

11.3.1
Shock waves are generally formed when the source of sound waves is travelling above the speed of sound. As the plane (since it's usually a plane) approaches the speed of sound, the sound waves don't really get away from the plane, but rather build up in front of the plane. Over time, many of these waves constructively interfere, producing what is known as the sound barrier. Once the plane moves faster than this, the sound waves are left behind the plane, creating a shock wave, which follows under the plane. The angle of the shock wave can be found by taking one point to be the source, then finding where the source would have been 1 second ago. From this point, calculate how far the wave would have gone out from this point in that second, and draw in the circle. A line can then be drawn from the point to the edge of the circle (in a tangent). This will be at 90 degrees to a line from the center, and since two sides are known, the angle of the shock wave can be calculated.

11.3.2
Doppler effect : This effect is seen in the change in frequency of sound when either the source or the observer are moving. This, therefore, affects the actual number of waves the observer hears per second, and so changes the observed frequency. If the observer and source are moving closer together, then more wave fronts will be observed per second, and so the frequency will be higher. If they are moving apart, then fewer wave fronts will be observed, and so the frequency will be lower.

11.3.3
When the source is at rest, the distance between wave crests is λ. If the frequency is f, then the time (T) between crests is 1/T (=f). If we then assume that the source is moving towards the observer at vs, then in time T, the first crest has moved a distance (d) where d = vT. In the same time, the source has moved ds = vsT in the same direction. At time T, the source emits another wave, and so the distance between these two will be d - ds ... Therefore, the new wavelength will be d - ds. This can be expressed as follows.

λ` = d - ds ( and since d = λ, and ds = vsT)

λ` = λ - ds vsT

λ` = λ - vs x λ/v

λ` = λ ( 1 - Vs/V )

The new frequency therefore is given by the following

f` = <SUP>V</SUP>/<SUB>λ` = v / ( λ x ( 1 - Vs/V ) ), and since V/λ = f

f` = f / ( 1 - Vs/V ) (which is the equation in the data book). If the motion is away from the observer, then Vs will be negative, making the sign in the middle positive, but this can be determined as you work out the problem if you know whether the wavelength should be higher or lower.

When the observer is moving towards the source, the problem is slightly different because the wavelength isn't actually changing, but rather the relative velocity of the waves. The speed of the wave, V` = V + V<SUB>0</SUB> where V is the velocity of sound in air. Thus, f` = V`/λ = V + V0/λ. Since λ = v/f, we get the following.

f` = ( 1 + Vo/V ) f. This is for an observer moving towards the source, a sign change will be necessary as above.

These can both be applied as appropriate to solve problems.

11.4.1
Unsure about this one. Please fill us in if you know or have somewhere to look it up.

11.4.2
An overall graph of a standing wave will look like a sine curve superimposed over a -sine curve. At any given point in time, though, consecutive anti-nodes will be on opposite sides, so if one is up, the next will be down, then up and so on. The nodes will divide the string into equal segments, and so calculations can be done with a sort of arithmetic sequence thing.

11.4.3
Equation relating fundamental frequency to tension and mass per unit length.

[mailto:eheddle@hotmail.com Edward Heddle] tells us that we've confused the symbols for tension and period here.

"The formula for the speed of a wave in a string is v = (T/µ)^(1/2), where T is the tension (N) in the string, and the linear density µ = mass/unit length (kg/m). This can be shown with dimensional analysis. This v can be combined with the formula v = fÎ. (N.B. T = 1/f is not the same as tension.) Fiddling around with 2l = Î, gives the fundamental as f(1) = 1/2l x (T/µ)^(1/2)."

(Originally we had the following) First, I should mention the equation v = √(T/µ). This allows us to calculate the velocity of a wave in a given string based on T, the period and µ, the mass per meter of string. This equation can be equated to v = f x λ. We can then play around with it to get various formulae, for example, 1/µ = f3 x λ2.

11.4.4
As noted previously, an open end in a pipe will have an antinode, and a closed end will have an node. Therefore, a closed-closed pipe will have a half wavelength, as will an open-open pipe, but an open-closed pipe will have one quarter. These are the fundamental frequencies, then half wavelengths can be added to get the first, then second and so on harmonics. Most of the problems involve relating the length to the wavelength or frequency of the sound produced.

title: IB Physics/Wave Phenomena


text:
IB Physics/Electromagnetic Induction

12.1.1
Coulomb's Law : F = Q1Q2 / 4 x π x Eo x r2. Q is in coulombs, r in meters. The force is dependent on the two charges and the square of the distance between them. Opposite charges attract and like charges repel. Vector addition of these forces may be necessary in some problems, but shouldn't present any major difficulties. The force on each of the two charges in question is opposite in direction, but equal in magnitude.

12.1.2
Electric field strength is equal to Force/charge. To find the field strength at a given point, put a point charge of 1 coulomb at the point, and then calculate the force on it. From there you can find the field strength and direction by vector addition.

12.1.3
Electrostatic potential : The work done (in joules) to move a charge from an infinite distance to a given point in the field, divided by the charge (in coulombs). In equation form, V = W/q. The SI unit is volts (where a volt is a joule per coulomb). This value is a scalar, so it doesn't matter what path you take between two points.

In a radial field, V = 1/4 x π x Eo x q/r. Note, this equation only applies as long as r is outside the conductor producing the field. If it's inside, then V = 0. In this case, q is the charge on the point creating the field. Both of these equations are in the data book.

12.1.4
1 electron volt is defined as the work done to move an electron through a potential difference of 1 volt. This can be plugged into the above equation, to show that 1eV = 1.6 x 10-19J.

12.1.5
The potential at a point in a field around a point charge or a uniformly charged sphere can be found with the above equation (V = 1/4 x π x Eo x q/r).

For a collection of point charges, the fact that electric potential is a scalar greatly simplifies the problem, because direction need not be considered. Simply add up all the different potentials from each point charge to find the total.

Inside a hollow sphere, the value of V will be zero until you're outside the sphere, at which point it jumps to a max, then falls away to zero at infinity.

12.1.6
For parallel plates, the equipotential lines run parallel between the plates, and then diverge out from either end.

For point charges, two opposite charges create circles of equipotential around them, but squashed in on the side closes to the other point. Eventually there will be a line running straight between them.

For like charges (maybe not necessary), the equipotential lines produce a sort of figure 8 with the center cut out, and each point charge in one of the 'holes' there are a bunch of these figure 8 shapes radiating out.

Between parallel plates, equipotential lines can be related to electric field strengths, because they evenly divide up the space between the plates, as does the field strength. Also, equipotential lines will always cross filed lines at 90o, so that should help find exactly where they go.

12.2.1
To find the force on a moving charge, or an electric current in a magnetic field :

Moving charge : Using the right hand palm rule, find the direction of the force (remembering we're talking about conventional current, so it goes forward for positive charges, backward for negative). Find the acute angle between the direction of the magnetic field at that point, and the direction in which the particle is traveling. Substitute into F = qVBsinØ to find the magnitude of the force (Ø is the angle, B is the field strength, V is the velocity and q is the charge).

Current carrying wire : Find the direction of current flow in the wire, and thus the direction of the force. Find the angle between the field lines and current flow. Substitute into F = IlBsinØ to find the magnitude.

Note that when Ø = 0o (i.e. when the motion is parallel to the field) there will be no force. Both of these equations are in the data book.

12.2.2
Galvanometer : A permanent magnet is set up around a loop of wire. This wire is allowed to rotate on a axis, but has a spring attached to always pull it pack to parallel when there is no force. When a current is passed through the loop, this causes a force on the loop, and it rotates. This pivoting moves a marker attached to the axis, and so shows the current which is flowing through, since the grater the current, the further it will turn before the force is equalized by the spring.

Loud speaker : A loud speaker consists of a metal coil attached to the stiffened cardboard of the speaker, with a permanent magnet surrounding the coil. When there is current running through the coil, this causes forces back and forth (as appropriate) on the coil, and thus it vibrates the cardboard. With the right alternating current, different frequency vibrations can be produced, thus producing sound.

Electromagnetic relay : The object of a relay is to complete a circuit when a current is passed through an electromagnet. This is done with a coil of wire which, when current is passed through it, pulls a piece of metal near it towards it. Since this metal is on a pivot, it forces a wire attached to it above up, and onto the other wire, thus completing the circuit. When the current in the coil is released, the spring action disconnects the circuit.

12.2.3
The magnetic field around a current carrying wire is defined by B = µo/2 x π x I/r. I is the current through the wire, and r is the distance away from the wire where the field is being measured.

For a solenoid at least 10 time as long as it is wide, the field inside is constant, and defined by B = µo x NI/l, where N is the number of loops of wire, l is the length, I is the current and µo is a constant. The field strength is equal anywhere inside the coil as mentioned above.

Both these equations are given in the data book.

12.2.4
The force between two parallel conductors is defined by F = µo/2π x I1 x I2 x l/r. Where r is the distance between them, I1 and I2 are the currents in each and l is the length of both wires. Force per meter can be found simply by substituting in 1. As can be seen, if the currents are in opposite directions, the force will be negative, or away from the other conductor.

The ampere is defined based on this (the current which produces a force of 2 x 10-7N between two wires 1 meter apart), since it is easy to accurately vary the current in a wire. The coulomb is then defined from this (charge in coulombs = current x time).

12.3.1
Magnetic flux is defined as Φ = BA (field strength x area). If the field is not perpendicular to the area in question, however, Φ = BAcosØ, where Ø is the acute angle between the field direction and the normal to the area. Flux linkage is basically the change in NØ, where N is the number of turns of wire, and Ø is the flux.

12.3.2
Faraday's Law is E = -N x ΔØ/Δt (the induced emf is equal to the number of loops x the change in flux over time).

Lenz's law says that an induced emf will always produce a current who's magnetic field opposes the original change in flux. Farraday's law says that the induced emf is proportional to the flux cut divided by the time taken.

12.3.3
emf = Blv (a wire of length l moving perpendicularly through a field of strength B at a velocity v).

This can be derived from F = ΔØ/Δt as follows :

F =

B x ΔA/Δt =

B x l x v x Δt/Δt =

Blv.

Or from F=IlB as follows :

(charge Q)(Voltage) = Work done V = Work Done / Q = Force x distance / Q = IlB x s / Q = Q/tlBs / Q The Qs cancel out and we're left with V = lB s / t = Blv.

12.3.4
(I assume what we're talking about here is a loop of wire rotating in a magnetic field.)

We take a coil with N turns of wire, and an area A rotating in a field of strength B at an angular velocity of w. The flux linkage at any given point is BAN x sin wt, where t is the time, as it rotates, starting from to when the entire coil is parallel to the field, and so produces no emf.

We then play with some calculus (substitute it into Neumann's equation, N being already accounted for, and do it like a calculus equation) to get emf = BANw cos wt. This equation is not in the data book, so it might be worth remembering.

12.3.5
An AC generator is, just like above, a coil being forced to rotate in a magnetic field. This produces an alternating current because each half turn, the effective orientation of the coil is reversed (the side that was going left is going right), and so the current is also reversed. The two ends of the loop are connected to slip rings which are allowed to turn, and brushes rubbing on them, and running the alternating current out to the rest of the circuit.

12.3.6
Average power consumption = IrmsVrms and Irms = Io/√2 (same for Vrms). This equation is in the data book, and can be derived as follows.

P = IoVo x sin2wt, thus Pav = 1/2 IoVo and so the above rms bits can be found. Vrms = Vo/√2.

12.4.1
A cathode ray oscilloscope (CRO) is a tool for measuring variations in current from a source. The CRO provides a continual graph of the current over time on the screen. It's difficult to describe how exactly to use one, but hopefully you'll have to have tried it.

At the back of the CRO there is an electron 'gun' where a beam of electrons are produced by a large potential difference between an anode and cathode (the 'ray' comes off the cathode, making it a cathode ray). It then passes through two sets of perpendicular deflectors. The horizontal one is controlled by the settings of the CRO, and makes the light trace from left to right (and then jump back to the start). The vertical one is indirectly controlled by the source current, and produces the up and down sine type curves.

The difference between this and a TV tube is that a TV strikes every pixel with the cathode ray once per frame (and may have multiple pixels with different colours) The CRO, however, strikes in a sine curve through each pass. Other than that, they both work on the same principle.

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

title: IB Physics/Electromagnetic Induction


text:
IB Physics/Quantum and Nuclear Physics

13.1.1
Millikan's oil drop experiment involved first producing small droplets of oil with an atomiser. Some of these then fell through a small hole, and into a region between two fields. Using a variable resistance, the strength of the field was adjusted until the upward force of the field (the positive plate was above, negative below, but could be reversed for drops of opposite charge) equals the downward force of gravity. After it was balanced, the potential difference was recorded, and then the drop was allowed to fall. After falling some distance, the drop falls at a constant speed (where the force of gravity is equaled by the air resistance). This terminal speed is measured, and allows the mass to be found using Stoke's law.

Then, we can equate Fg and Ff, as follows : qE = mg therefore q = mg/E.

Since the mass of the drop can be found, and both g and E are known, we can find the charge of the drop. By graphing these charge values, we can find that the smallest difference between them is e, 1.6 x 10-19C.

13.1.2
Thompson's experiment is based around the counteracting effects of electric and magnetic fields.

First, the two fields are adjusted so an electron beam passes undeflected between them, meaning the deflection effects of each cancel out. Therefore, Fe = Fb, and so Eq = Bqv. From this, v = E/B.

The electric field is then removed, meaning the beam is deflected by the magnetic field. Since the force from a magnetic field = evB, from Fc = mv2 / r we get evB = mv2 / r. This simplifies down to e/m= v/Br. We can measure the radius of curvature, we know the velocity from above, and we know the strength of the magnetic field (B), and so the value of e/mcan be found.

13.1.3
The distance of closest approach of a particle to the nucleus can be found by the conservation of energy. The initial energy of the particle is defined by Ek = 1/2mv2. This energy is converted into potential energy as the charge approaches the nucleus. The potential energy at a given point is equal to the work done to move the charge to that point, so W = qαV. V for a radial field is 1 / 4 x π x Eo x qnucleus / d, so the potential energy is qα qnucleus / 4 x π x Eo x d. When this is equated with the kinetic energy, all the terms are known except the distance, which can therefore be found.

13.2.1
A mass spectrometer works as follows : First we start with a source of ions, all with a +1 charge. These are accelerated through an electric field. These ions then enter a velocity selector, an area with both magnetic and electric fields applying a force in opposite directions, so as to cancel each other out for a particular mass. As above the two forces equal out, and we get Fe = Fb, and so Eq = Bqv, therefore v = E/B. Thus, only ions with a particular speed are allowed through to the next stage.

The electric field ends, and the ions are deflected through a magnetic field, resulting in a circular path. This means there is a centripetal force supplied by the field, and so mv2 / r = Bqv, which rearranges to m = Bqr/v. Assuming the B field is varied to keep the radius constant, mass is proportional to magnetic field strength, as everything else is constant.

13.2.2
Chadwick discovered the neutron by bombardment of Beryllium with alpha particles. A wax block (with its chain of hydrocarbons it serves as a proton source) is placed after a short distance, and protons are detected after the wax block. Thus there must be some way to 'knock out' the proton. From the energy the protons are in, the Compton effect due to possible gamma ray emission is too low to compensate. If the particle that alpha knocked out is a neutron though, it fits perfectly both by the conservation of energy and the conservation of momentum (since the proton and neutron have approximately same mass, we would expect the 'pool players' result') yet these particles are indeed uncharged. Further more, when these are to collide with protons in cloud chambers, a right angle track would result.

13.2.3
The mechanisms for beta decay :
 * p -> n + e+ + a neutrino (Greek letter nu, lowercase) and

 * n -> p + e- + an antineutrino (Greek letter nu, lowercase, with a bar on top)

13.2.4
For decay chains, everything can be written as a series of nuclear equations as in the following example.

AZX + 0-1e -> AZ-1Y. Note that the totals must be kept constant on both sides, but that should be fairly easy.

13.2.5
The equation N = No x e-λ x t is in the data book, while ΔN/Δt = -λ x N is not.

Both those equations can be used to find decay information. N is the number of radioactive nuclei, No is the starting number and λ is defined as ln2/half life (that's in the data book). ΔN/Δt is the rate of decay.

13.2.6
Deduce λ = ln2/half life as follows.

We start by taking a time after one half life, therefore N = No/2. N = No x e-λ x t becomes 2 = eλ x t. We then take the natural logarithm of both sides to get ln2 = λ x t0.5, and this rearranges to the required expression.

13.2.7
By Coulomb's law, the protons in the nucleus should all repel each other and break it to bits. Clearly there must be another force holding it all together, which we call the strong nuclear force. This is a force which greatly outweighs the electromagnetic repulsion, but only acts over a very small distance (within the nucleus).

13.3.1
Atomic emission and absorption spectra result from the fact that electrons can move between energy levels when they have sufficient energy put in. They will then fall back, emitting a defined amount of energy as light. Emission spectra result from electrons being excited by electricity (or something), then emitting light as the electrons fall back. Absorption spectra result from electrons absorbing energy from electromagnetic radiation, and so effectively blocking it. In the emission case, there will be a series of thin bands representing the wavelengths of light produced, and for absorption, there will be a full spectrum with some lines cut out (the wavelengths that were absorbed).

13.3.2
Bohrs postulates :
 * 1) Some stable orbits exists (assumed circular)
 * 2) Electrons absorb/emit energy in changing orbits
 * 3) Quantum condition (rules for changing orbit) : $$mvr = n\frac{h}{2\pi},\ where\ n=1,2,3...$$

The last formula can be derived by equating Angular momentum L with $$n\hbar$$ where $$\hbar = \frac{h}{2\pi}$$ (pronounced h-bar). Historically it is based on experimental discovery. We'll call this equation *1 in subsequent sections.

13.3.3
Describe the spectrum of atomic hydrogen and account for it, using Bohrs' model. (This account should be slightly more detailed than that required by the IB.)

The charge in the nucleus = Ze Charge in electron = e Assuming a circular path, centripetal force = mv2 / radius. Since this force is supported by the electronic attraction, mv2 / r = k (Ze)(e) / r2 Simplifying we obtain r = k (Z)(e2) / m(v2). We'll call this equation *2

From above *1, mvr = n (hbar), thus rearranging we obtain v = n (hbar) / mr. Plugging into *2, simplify and you'll obtain a equation, which relates the orbital radius with (constant)(n2). Energy of the electron = KE + PE = 1 / 2 m(v2)+ k Z(e2) / r. Substitute v by expression *1, we'll arrive at E at nth shell = constant / r at nth shell. Since r is proportional to (n2), E = constant / n2. (required by IB)

Knowing this and E = hc / λ (Planck's equation), you'll also need to know the Rydberg equation. This is shown in the syllabus, and can be easily derived from the above.

13.3.4
Evaluate the success and limitations of Bohr's model

Success :
 * 1) Explains why atoms emit and successfully predict emission for Hydrogen.
 * 2) Explains why atoms absorb
 * 3) Ensures the stability of atoms
 * 4) Predicts accurately the ionisation energy for Hydrogen.

Limitations :
 * 1) Not successful for multi-electron atoms.
 * 2) Can't explain fine structures (emission lines existing as 2 or more close lines)
 * 3) Can't explain bonding of atoms in molecules or solids & liquids
 * 4) Can't explain different intensity of spectral lines.

13.4.1
Einstein's mass-energy equivalence : $$\Delta E = \Delta mc^2$$, E is in joules, m in kilograms, and c is the speed of light in a vacuum.

13.4.2
Unified mass unit : One twelfth the mass of a carbon 12 atom.

Mass defect : The amount of mass which is converted into energy in a nuclear reaction.

Binding energy : The energy equivalent of the difference in mass between the nucleus of an atom, and the masses of the individual protons and neutrons which make it up.

13.4.3
The binding energy can be calculated as described above, by finding the total mass defect between the individual nucleons, and the whole nucleus. The binding energy per nucleon is therefore, this divided by the number of nucleons.

13.4.4
The graph of atomic number vs binding energy per nucleon runs from Z=2 increasing rapidly (with 3 peaks I don't think we need to worry about) to about Z=20 where it becomes relatively flat at around 8 Mev per nucleon then begins to drop off after Z = 60. The higher an element is (i.e. the more binding energy it has) the more stable it will be, and so the most stable elements are those around Z = 20.

13.4.5
Fission is the process by which an atom breaks up into smaller fragments. This is often caused by the addition of neutrons to the atoms, causing it to become unstable and eventually break up. This breaking up may, in some cases, produce more neutrons, and so these can then go on to produce more fission reactions, creating a chain reaction which perpetuates itself.

13.4.6
Fission is good because it provides a lot of energy from a source that is more viable long term than fossil fuels, and because it is relatively clean in terms of air pollution compared to fossil fuels. The down side is that it produces radioactive material which must be stored somewhere, and also, that it can be dangerous if not controlled properly (i.e. meltdowns).

13.4.7
Nuclear fusion occurs when two smaller nuclei fuse together to form one bigger, and more stable nucleus, and produce lots of energy in the process. Initiation of fusion requires a great deal of heat, because the nuclei must be given enough initial energy to overcome the Coulomb repulsion between them as they approach. Energy calculations can be made using E = mc2, when the masses of the different fragments are given.

13.5.1
The explanation of the photoelectric effect is that the energy carried by light is broken into discrete units, the size of which depends on the frequency of the radiation. The energy carried in each 'photon' is defined as E = hf (plank's constant x frequency). The atoms require a certain amount of energy to release an electron, Wo = hfo, where Wo is called the work function. If there is more energy than this, then that may be given to the electron in the form of kinetic energy, and so E = hf = Wo + 1/2x m x vmax2.

13.5.2
The photoelectric effect can be measured by applying a stopping voltage in the opposite direction to the current induced by the photoelectron emission. As the frequency of the light is increased, more energy will be required to stop these electrons. If the frequency is decreased, however, there is eventually a point where no emissions occur, and so no voltage is required.

hf = hfo + eVs, where Vs is the stopping voltage.

13.5.3
X-rays are produced by first placing an anode and cathode in a vacuum tube. Behind the anode is some type of photo-sensitive material, and between the two is a potential difference of about 150,000v. The cathode is heated to produce thermo-electrons. These electrons the accelerate towards the anode. When the electrons are deflected, by coming close to the nuclei, their kinetic energy changes. This change results in the production of an X-ray.

Since the electrons may come close or stay far away from the nucleus, the x-ray spectrum is continuous, not discrete. There are, however, peaks caused by inner shell electrons being excited by small energy loss. These peaks occur on the left side of the curve, which is generally an inverted parabola type shape. There is a shortest wavelength possible for the x-rays due to the fact that when electrons lose all their kinetic energy, there is no way to further increase the wave frequency.

13.5.4
When electrons lose some of their energy, X-rays are produced. The short wavelength limit is when they lose all of it so eV = hc/λ = 1/2 mv2. Rearranging, we get λmin = hc/eV

Apparently there was some other stuff I meant to come back to, but no longer remember.

13.5.5
DeBroglie's equation is λ = h/p. From this, we can see that all mass has a wave equivalent, and any wave has a mass equivalent. p refers to the momentum of the particle, f its frequency, and λ its wavelength.

13.5.6
The velocity of an electron can be found from 1/2mv2 = eV, and this can then be used in the equation mv=h/λ to find the electron's wavelength. This can be seen or verified by the diffraction of electrons through thin crystals, showing that electrons have a wave nature.

13.6.1
Linear accelerators are designed based on a series of 'tubes' through which the particles are pulled, and then pushed by electric fields. The lengths of the tubes become longer and longer so the frequency with which the electric fields must oscillate is constant.

13.6.2
Circular particle accelerators work on the basis of magnetic fields making the particles rotate, and when they cross between the two Ds, they are accelerated between them by a electric field. The radius inside is defined by r = mv / Bq, and so as the velocity increases, the magnetic field must be increased to keep the radius constant.

13.6.3
When a particle having been accelerated collides with a fixed target, both usually break up into smaller fragments. These can sometimes be identified in a cloud chamber.

13.6.4
Particle anti-particle pairs are only really produced from interactions involving great amounts of energy. When two such particles collide, they completely annihilate, producing only energy.

13.6.5
Particle / anti-particle pairs :
 * electrons - positrons (positrons are uncommon).
 * proton - antiproton (these are the same except for charge)
 * photon - photon (same particle)

title: IB Physics/Quantum and Nuclear Physics


text:
IB Physics/History and Development of Physics HL

E.5.1
Carnot first developed the concept of entropy in his work on heat engines. By building a theoretical model of an engine which achieved 100% efficiency, he was, in fact, assuming that entropy could be made constant, as everything in this model was being done reversibly.

Clausius and Kelvin also worked on the macroscopic interpretations of entropy (Clausius coined the term).

Boltzmann developed the microscopic interpretation, based on the random motion of particles.

E.5.2
Clausius used entropy to describe the degradation, and unavailability of energy. Even though energy is conserved, it is only useful if it can be made to do something. For example, a system where one area is hot, while the other is cold is more useful than a system with the same amount of heat, but distributed evenly. When energy is spreads out more evenly, it is said to be degraded.

E.5.3
Maxwell's daemon is a 'thought experiment' which tries to overcome the law that entropy must always increase. We imagine that there are two containers filled with gases, connected by a small pipe. In the centre is a daemon, and, as daemons do, it wants to try to increase the temperature in one side, and decrease that in the other, thus defying the law of entropy. He/she/it does this by opening and closing a door, allowing fast moving particles to go through into the left, and slow moving particles into the right. This separation of fast and slow moving particles means entropy is being defied.

The problem with this is that for the demon to be doing this, energy must be used, so even if the entropy inside is decreasing, the energy to achieve this will be greater, and so there will be a larger increase in entropy somewhere else.

E.5.4
When analyzing systems in thermodynamic terms, it is important to consider both the first and second laws. Both the quantity of energy, and the quality must be considered to understand the usefulness and ability of the system to do work.

E.6.1
The cavity radiator oscillator (aka black body radiator) relates to objects which absorb all the electromagnetic energy which strikes them. As the wavelength of the radiation decreases ( i.e. increased frequency ) the oscillations of the particles in the object increase. It therefore stands to reason that, eventually, the oscillations would become so large that the radiator would simply fall apart. This wavelength was calculated to be within the ultraviolet spectrum, hence the name ultraviolet catastrophe.

E.6.2
Max plank's idea was that energy could only be absorbed or emitted by radiators in discrete amounts, and so a low frequency is less likely to give energy to a high frequency oscillator because the energy emitted by such a low frequency oscillator would not be of sufficient energy. Thus, the ultraviolet catastrophe does not occur.

E.7.1
The four fundamental forces are:

Gravity : Long range, very weak compared to the others.

Electromagnetic : Long range, stronger, but only significant at small distances.

Weak : Short range, and stronger than gravity, weaker than electromagnetic.

Strong : Very short range (only affects the nearest nucleons), and very strong.

E.7.2
The fundamental particles known in 1935 were protons, neutrons and electrons.

E.7.3
The strong force acts to hold the nucleus together.

The weak force is involved in the transformation of neutrons into protons by beta decay.

E.7.4
The model used for the movement of forces is that of a 'virtual exchange' particle. Each of the 4 forces has an associated virtual particle (though the graviton's existence has never been proved).

 * Strong force : gluons (formerly mesons, which is what the syllabus says)
 * Electromagnetic force : Photon
 * Weak nuclear force : Boson
 * Gravity : Graviton

A feynman diagram basically shows the two particles Moving towards each other, then moving away, with a dotted line between the turning point representing the movement of the virtual particle.

E.7.5
Things that are conserved in nuclear reactions :
 * Mass-energy
 * Momentum
 * Angular momentum
 * Electric charge
 * Lepton number
 * Muon number
 * Baryon number

These should all be considered when nuclear reactions are being written (but the momentum ones can't really be written).

E.7.6
A symmetry operation (thank god for abstract algebra) is an 'operation' which involves a rotation or translation in which everything is conserved.

E.7.7
Murray Gell-Mann's 'eightfold way' was a model based on the SU3 group which was designed to model the particles which quarks composed. It was used to predict the existence of the Omiga minus particle, and the properties thereof before it was physically found.

E.7.8
There are six quarks : up, down, strange, charmed, top (aka truth) and bottom (aka beauty). In addition to these, each has its own antiquark, with opposite charge, and opposite strangeness, charm, topness or bottomness as appropriate. They also have a baryon number of -1/3 as opposed to 1/3 for the normal quarks.

These quarks are composed together to form the different particles. They each have charges of ±1/3 or ±2/3, and get stuck together to form particles like the proton (uud) and neutron (udd). Up quarks have a charge of +2/3, down quarks have a charge of -1/3 (note that the other types are not as common).

E.7.9
Quarks also have a quality called color with possible values of red, blue, green, anti-red, anti-blue and anti-green.

The idea was originally proposed to preserve the Pauli exclusion principle. The quarks are held together by the exchange of particles called gluons (which is sort of like the strong force). There are 8 gluons, 6 of which have a color charge.

E.7.10
Organizational principles in science : They're good, and help to, well, organise things. They help to show trends in data which otherwise is just a mess.

Examples are the eightfold way, and the unification of the forces.


text:
IB Physics/Astrophysics HL

Thanks to Tímea Garlati and a little help from Mephistophyles

F.2 Stellar radiation and stellar types
=Binary Stars= A binary star system is the orbiting of two stars around a common center. There are three classes of binary stars: visual, eclipsing, and spectroscopic.

Visual binaries
Appear as two separate stars as they orbit their common center. The center is know to be the mass of the two stars. The common period of rotation can be found with the equation

T²=4π²d³/G(M1+M2)

where d stands for the distance between the stars. Note that these stars will always remain diametrically opposite of each other. This is due to the gravity each star produces and contains.

Eclipsing binaries
This binary system occurs when the stars are oriented to a degree that when viewed from earth, one star will appear to stand in front of the second star. This will result in an eclipse, which may be total or partial. Depending on the stars, the light will bend around it, which will create a light curve pattern. This can be measured by the temporary increase of the luminosity, as the stats light emission is distorted.

Spectroscopic binaries
This binary system can only be defined if the light from one or both of the stars is observed as suffering from the Doppler effect. This shifting will cause the star to fall under two categories: the blue shift, which means moving towards the earth, and redshift, meaning moving away from the earth. The change is determined by

z=[λ-λ0/λ0]

were z represents the shift, and y is the wavelength of the spectral line and y is the wavelength earth observes.

In addition, the stars relatives speed in contrast to that of light will illustrate that there is a correlation between the shift and the speed of the star.

E.H. and J.N.

The Olbers 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.

/M.H.

The Development of the Universe
There is still a relatively large argument about how the universe actually came to be. In physics, the development reaches one conclusion through the Big Bang theory, but new theories are in speculation. The expansion of the universe can be described in mathematical terms. Consider

x(f)=R(t)x0

if the distance between two galaxies was once x0, their separation after some time would described by (t).

Function R(t) is known as the scale factor of the universe. The scale factor could be loosely interpreted as the radius of the universe.

While the function is convenient enough, it leaves scientists with a large obstacle. The challenge of finding what the scale factor actually comes to leaves only three possibilities. They work in relation with the laws of general relativity.

[clearly not finished yo] E.H. and J.N.

F.5.1
The longer the period of light variation of a Cepheid, the greater is the luminosity. Observing the luminosity and the period the absolute magnitude can be determined. Comparing the absolute luminosity to the apparent luminosity we can calculate the distance to the Cepheid and surrounding stars. They serve therefore as 'distance markers' for distant stars where parallax can not be applied.

E.H. and J.N.

F.6 The expanding universe
Movement of Constellations/Stars over night/year

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.

www.astro.psu.edu/users/rbc/a1/lec2_2d.html

F.7.1
'No observer can determine by experiment whether he or she is accelerating or is rather in a gravitational field.' For example, a pilot in fog, while making a steep turn can not tell whether he is accelerating by gravity or by the plane's thrust without looking at his instruments.

inertial mass = gravitational mass.

F.7.2
The effect of gravity upon time can be visualized as a stretched rubber-sheet, representing the 3 dimensions of space and one of time that is deformed by gravity. The more massive the object, the greater the deformation.

Does it mean that we don't know whether we are accelerated through time or pulled backwards by gravity?

F.7.3
The effect of gravity upon time can be visualized as a stretched rubber-sheet, representing the 3 dimensions of space and one of time that is deformed by gravity. The more massive the object, the greater the deformation.

Does it mean that we don't know whether we are accelerated through time or pulled backwards by gravity?

EDIT: It is actually the fabric of space-time that can be visualized as a rubber-sheet. When massive celestial bodies such as planets are placed on this fabric, they bend it inwards. This curvature on the fabric can be defined as the effect of gravity. When smaller objects go into this curvature, they start spiraling around the central planet and get closer to the core as they sprial. You can visualize this motion as the water that spirals down the drain.

Therefore, we can say that gravity causes acceleration on the space-time fabric.

F.7.5
Light is bent by strong gravitational fields. This was observed when light from distant star could be observed although the Sun stood in its way.

F.8.1
Stars undergo changes as their hydrogen is used up, first off they try to fuse the heavier elements, but this only works under certains circumstances and for a finite time. They either red giants after they go supernova, or they release a planetary nebula (nothing to do with planets, another form of expansion not as bad as going supernova). After going supernova stars become red giants, due to their large masses, or they decay into a neutron star. If they release a planetary nebula, they become a white dwarf or a brown dwarf, all depending on their mass.

(See also F.1.1)

F.8.2
Clouds of hydrogen and helium forms into 'Main sequence' stars.

Nuclear fusion takes place: H + H -> He, then He + He -> Be.

Further fusion takes only place in heavier stars, otherwise the pull of gravity forces the star to contract and cool to a red dwarf. If further fusion takes place the star becomes a red giant.

Small mass red giants : 1.4 solar masses (Chandraseka limit) can not withstand the pull of gravity, so it shrinks, becomes extremely hot, till it finally cools into a white dwarf.

Large red giants : Fuse until the formation of iron, but thereafter no fusion can take place without energy addition. So the star contracts, and heats up because of the large KE in the particles. When it can't be compressed further it explodes in a supernova. Such a great explosion may leave behind a very dense star made up of mostly neutrons, i.e. a neutron star.

Lighter neutron stars : With solar masses below 2-3 solar masses are thought to form Black holes because the gravitational force, which increases with mass, doesn't even allow light to leave the surface. Note, it is not a hole as such just a small extremely dense mass.

F.8.3
Pulsars are neutron stars that radiate energy at regular periods (See F.1.1)

Quasars suggest the existence of black holes, since the accelerated matter that black holes draw in could release its energy in the form large amounts of light. X-ray is possibly produced as the accelerated matter due to Black holes is compressed and heated to millions of degrees.

F.8.4
Since black holes are really massive the 'stretched rubber sheet' of space-time becomes very deformed.

The Schwarzhild equation tells that Rs = 2GM/c2 meaning that there's a critical radius when a mass becomes a black hole. (G is the gravitational constant, M is mass and c is the speed of light). For Earth this radius is 1cm.

The event horizon is the boundary at which light can not escape from the black hole. The singularity is a spot in the middle of the event horizon to which the mass shrinks, since no known force can stop the contraction. Here all time, space, matter and energy end.

Wormholes should allow us to disregard the curvature of space-time and let us travel through space instantaneously.

Not exactly; wormholes allow us to travel faster between two points than would be possible over the plane of space-time. The problem with relativity in it is that it views space-time as a flat plane with holes or dents that represent large masses. But for wormholes to effectively work, space-time needs to be curved or even spherical (like a soccer ball). Also, if you appear on the other end faster than light could travel all the way around, you are effectively violating (in some complicated way) causality.

But how does this link to general relativity

General relativity states space-time curves because of mass, yadda yadda. A singularity has such high mass that it can curve space time in on itself creating wormhole. Or it can cause a tear in the continuum, into which everything in the event horizon is eventually pulled, out of the universe itself.

not exactly, black holes have very high density areas called singularities, they are very curved, but they don't have infinite mass, but high enough to curve light into it, the event horizon is found using The Schwarzhild equation (see above). The wormhole thing is something entirely different, it's not a rip in the fabric, more like a tunnel between two black holes. Einstein did some work on this and it led to a recent discovery at CalTech where they found it was theoretically possible to time travel, but you need to be REALLY small and REALLY fast. Anyhow, that should cover that, any more questions, just post them. The only thing on the syllabus as far as I know is the Schwarzhild equation.

title: IB Physics/Astrophysics HL


text:
IB Physics/Optics HL

H.6.1
Circular mirrors do not focus all their light to a single point (that's what parabolas do). Close to the center, the shape of a circle and a parabola (on it's side) are very similar, and so paraxial rays (close to the center) will be focused to a point, but the further away from the center we go, the further away from the original focus the rays become (they come closer to the mirror as we move further out). This effect is known as spherical aberration, and causes a blurry image. It can be overcome with parabolic mirrors, but these are much more expensive to produce.

H.6.2
Optical fibres are (as mentioned in the SL section) composed of an optically dense core surrounded by a less dense coating. They rely on total internal reflection, and as such there is a critical angle beyond which the light will escape from the fibre. This angle can be calculated with the following equation.

sin Ø = ± √( n12 - n22 )

n1 is the refractive index of the core, and n2 the index of the coating. Hopefully that's obvious because the number we're taking the root of needs to be positive. This equation is in the data book.

H.6.3
Rainbows occur when the sun is behind the viewer, and there are raindrops in front. The sunlight enters the raindrops, and is totally internally reflected on the back, and then comes out and towards the viewer. The light is also dispersed within the rain drops, resulting in red being aimed the lowest, and violet the highest.

To draw the diagram, two drops must be drawn as large circles, one somewhat above the other. Sunlight enters at the same angle into each, and is immediately split into red and violet rays, where the violet ray is diffracted more towards the normal than the red ray. The rays are totally internally reflected by different angles from the curved back of the drop, and then cross over before leaving with red aimed down the most and violet up the most. (Note: this is the reverse order compared with the diffraction through a triangular prism.) From the top drop, the red beam should reach the viewpoint (illustrated by an eye) and violet from the bottom drop. These two rays are then extended back from the drops to produce a virtual image of the rainbow behind the drops (with red at the top, and violet at the bottom, and orange, yellow, green blue and indigo between).

H.6.4
The lens makers equation is 1/f = (n-1)(1/R1 + 1/R2).

f is the focal length of the lens, n is the refractive index of the lens material. R1 is the radius of the front side, and R2 the radius of the back (these radii are negative if the lens is concave on that side, positive if it's convex).

H.6.5
A large aperture lens will produce spherical aberration as a result of it's large diameter. This will cause the resultant image to focus, not at a single point, but rather over a line, which may cause problems in optical instruments.

Chromatic aberration results from the dispersion effect when light changes media. Because the lens has a large aperture, the light travels a greater distance in the glass, perspex or whatever, and so this dispersion becomes more significant than in a thin lens.

Both of these can be greatly reduced by the use of two lenses (ie a convex lens followed by a concave one).

H.6.6
How thin lenses can rectify some eye problems:

Myopia (Nearsightedness) : This is when an eye which can only focus on close objects. Parallel rays entering the eye focus before the back of the eye, crossing over as thus producing a blurry image. This can be rectified by using a diverging lens (a concave one) as this makes the light diverging rather than parallel as it strikes the lens of the eye, and so it focuses further back.

Hypermetropia (Farsightedness) : This is when an eye which is unable to focus on close objects, as it focuses too far back, the rays crossing behind the back of the eye. This can be corrected by using a converging lens (convex) which brings light rays from a close object closer to parallel to the eye's lens can focus them on the back of the eye.

Presbyopia : This is the decreasing ability of the eye to focus on close objects as the eye ages, and so it can be corrected in the same way as Hypermetropia above.

H.7.1
Two things limit the resolution of an image produced by a lens, aberration and diffraction. The aberration, discussed above, is ignored for the Rayleigh criterion. The Rayleigh criterion basically says that two images are just separable when the central peak of one is at the first minimum of the other (or further away from it). The smallest angular separation of two objects in the distance which can be resolved by a given lens (assuming no aberration) can be calculated with the following two formulas.

λ = asinØ where λ is the wavelength of the light being used, and a is the aperture of the (in this case) slit and Ø is the angular separation.

The more common one is for a spherical lens Ø = (1.22 x λ)/D where D is the diameter of the lens (in a multi lens system, for the objective lens).

Note the angular separation is in radians. π radians = 180 degrees.

The second equation is given in the data book and the first is just the same as the equation for calculating the position of the first minimum in single slit diffraction.

H.7.2
Resolving power : This is generally used for microscopes, where the object is placed close to the focal point of the lens, and is used to calculate the minimum distance between two objects for them to be resolvable. RP = s = fØ (where f is the lens focal length, and s is the minimum distance).

This equation is not in the data book. Resolving power can also be expressed as (1.22 x λ)/(2 sin a) where a is the angle formed if you take a line from the centre of the lens along the principal axis to the focal point, and then up to the top of the lens.

H.8.1
Linear, circular and elliptical states of polarisation, and Malus' law.

Malus' Law : I = Iocos2Ø. This law relates the intensity of light to the difference in angle between two polaroids. If the two polaroids are aligned so they both polarise in exactly the same direction, then there is an angle of zero between them, thus cos2 0 = 1, and the initial intensity, Io = I, the resultant intensity.

This relation ship can be derived by considering the vector components of the amplitude of the light waves. First, against a horizontal and vertical axis, draw a line going up and to the right from the origin. The angle Ø is the angle between this line, and the vertical axis. If we then take the length of the diagonal line to be Ao, then simple trig shows us that the vertical component is AocosØ. Since intensity is proportional to the square of amplitude, we get the relationship I = Iocos2Ø.

We need someone to volunteer to dig up some information about the states of polarisation.

H.8.2
Retarding plates: Can someone fill this in?

H.9.1
Stimulated emission is characterised by the fact that is produces highly focused, coherent and monochromatic light. Spontaneous emission, however, produces light in all direction (so intensity decreases rapidly) and is not coherent. Light going in different directions may be completely out of phase (or not, you just don't know).

Stimulated emissions generally occur when electrons within atoms are excited to a higher electron shell, and then fall back of their own accord (usually very quickly). Light stimulated emission occurs when an atom which is already in an excited state is hit by a photon of exactly the same energy. This causes the electrons to immediately fall beck to its ground state, and produce a photon. Thus, we have two photons, the original one and the produced one, and they are both exactly in phase, and travelling in the same direction. Under normal circumstances this would be very rare, but in the production of laser light it is intentionally achieved.

Production of laser (Light Amplification by Stimulated Emission) light : As seen above, focused, coherent light can be produced by stimulated emission, however this is very rare in normal circumstances. For a laser to operate, the atoms within it must be an inverted population (there must be more atoms in the excited state than the ground state). This excited state must also be a metastable state, which means the electrons stay in this excited position for longer than usual before spontaneously falling back to the ground state. How the metastable inverted population is actually achieved varies depending on the type of laser (and I assume it's not required here, but it's usually done with flashes of light or an electric current). The actual device consists of these atoms contained between two mirrors at either end of a tube (the sides of which absorb light). The mirror at the end which you want the laser to come out of is partially transparent, but the majority of light hitting it will be reflected. After a while (actually a very short time), some of the electrons start falling back to the ground state, and thus produce random photons in any direction. Eventually, one of these will be travelling along the tube, and strike an excited atom, producing two photons. These two will strike two more atoms, producing 4 and so on, thus the photons bounce back and forward between the mirrors continually increasing in number. Some of the photons leak out the partially transparent mirror and leave to form the actual 'beam' of the laser.

The importance of lasers in optics stems primarily from the fact that they can be far more precisely controlled than ordinary light. Thus they are far more precise to be used in calculations. Because they are monochromatic, their can be no chromatic aberration, and because they are highly focused, they produce very sharp points rather than a general dot you might get with ordinary light. In addition to this, they do not lose intensity even over long distances, and so they are useful for taking measurements over long distances.

H.10.1
Different methods for producing holograms, and seeing them in coherent light (are we referring there to laser light, or to light from a coherent point source ??? )

Producing holograms (basically a duplicate of the SL section)

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. The only variation I know of on this method is to use, rather than flat film, a thick emulsion. In this emulsion, the interference pattern is recorded in 3D, producing what is known as a volume hologram, which can be seen in white light (though best with a single, small point source).

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. Volume holograms works similarly, bit it is possible the use 3 different 'coloured' lasers, red, blue and green to produce a hologram which can then be seen in full colour under white light.

title: IB Physics/Optics HL


text:
Talk:Directing Technology/Evaluate

Please add your comment or suggestion in the following sessions.

Recommend & Report
Paige, thank you for changing the underline headings to italic. - Pei

Hi Pei, it seems that the wikibook doesn't pop up a new window after one click. The Wikibook reader can get a new window/tab by right-click. To get a-click-in-new-window feature, we need embed a piece of JavaScript code in the wikibook webpage. But, I found those JavaScript codes didn't work in this editing tool. --Lifang 02:36, 28 November 2007 (UTC)

I am still trying to figure out what to put in my section. It is hard. I will take a look at some districts technology evaluation reports. I will also put some chart and survey form in the section later on. - Pei

I've fleshed out a bit of the first part of my section and I'm pleased with what I have so far. I know I'll need to add some more detail but I have the flu right now and all I want to do is go home and go to bed. Paige H.

Do you figure out how to add charts/forms/tables/images to it? - Pei

''
 * ''Pei, I poached the html from another page and adapted it for my purposes. It was easier than trying to figure the whole thing myself. I wrote in most of what I want to include but I still need to go over it again to revise it.

Bill, I have a question about the wikibook. I just checked our chapter. You have moved your section "Recommend and Report" under my section "Collect and Analyze Data" (2.3 Recommend and Report). Your section should start from subtopic 3. I am not sure if you did this on purpose? I didn't want to change it. Pleas let me know. - Pei 11/19/07

(Pchu 06:56, 20 November 2007 (UTC)=Pei) Does anyone know how to create a link that opens a new browser?

 * After lots of discussion, we decided to leave our page as one long page. We thought about setting it up a one main page and 3 separate smaller pages but couldn't decide where to separate it. We also realized that Dr. Garrigan was talking about us when he mentioned the underlined titles and will be changing this as well. Any other input is welcome.

 * Part II- Just finished a big editing session. I added a reference, got rid of the underlines and changed them to italics and added a couple of links to other web pages. Whew!

Paige 11/28/2007

title: Talk:Directing Technology/Evaluate


text:
IB Physics/History and Development of Physics SL

E.1.1
Aristotle's model of the universe was that the universe was finite, spherical and above it was heaven. Everything was divided into 4 elements: earth, water, air and fire (the fire bun). Everything below the orbit of the moon was the earth, everything above the ether. The ether moved independently of the earth, which is a nice way to explain everything without thinking about it. Aristotle believed everything moved in circular paths, and that the earth was at the center of the universe.

E.1.2
Ptolemy's model was that 'heavenly bodies', such as the sun, moon planets and stars, orbited around the earth in circles, but there were little circles within these, to account for the fact that they didn't appear to match circles (called epicycles). There were eight levels in his model, with the outermost being like a fixed roof of stars.

E.1.3
Aristachis'/Copernicus' model had the earth orbiting the sun, along with the other planets, but the moon orbiting the earth. The stars were still like a fixed roof, out beyond the planets. Everything orbited in circles.

E.1.4
Objections to Copernicus' model were based primarily on religious issues, and enforced due to the close ties between the church and government. Galileo defended the model, with observations of moons orbiting Jupiter, however the church eventually forced him to publicly withdraw his support for the model.

E.1.5
Tycho Brahe's model had the planets orbiting the sun, but the sun orbiting the earth, and was intended to be like a compromise between the church and Copernicus' model.

E.1.6
Kepler's model was an extension of Copernicus', but had simple mathematical laws to describe the motions of the planets.
 * 1) The planets orbited on elliptical paths, with the sun at the focus.
 * 2) The planets orbited a segment of equal area in equal amounts of time, which meant when they were closer to the sun, they moved faster.
 * 3) T2 is proportional to R3. (T is period, R is radius). This means T12/T22 = R13/R23.

E.1.7
Newton's law of universal gravitation gave a mathematical, and theoretical basis for Kepler's model.

F = Gm1m2/r2, and F = mv2/r and since v=rw, F = mrw2. Equating the two, we get the following.

Gm1m2/r2 = mrw2, and w = 2π / T, so Gm1m2/r3 = m2π / T2, so T2 is proportional to R3.

Gravity provides the centripetal force necessary for Kepler's third law.

E.1.8
Simple models are considered as better than complex ones. Therefore the Copernican model was good, because it was simple, and allowed predictability (although it wasn't really all that accurate).

One other thing, since it was in the spec paper. Retrograde motion was the name for the observation that the planets did not move in predictable paths in relation to the stars, and even reversed directions, due to the relative motion of the earth.

E.2.1
Aristotle : To move an object, a force was required. To keep an object moving also required a force. Thus, the natural state of objects is at rest (there was no concept of friction).

E.2.2
Galileo examined the motion of objects on ramps. If an object rolled down one, then up another, then it always ended at the original height, irrespective of the slopes (because friction was minimal). This lead to the conclusion that if an object rolled down a slope and along a flat surface, it should go on to infinity, as it would never reach it's original height. This was also important for the fact that he was using experiment, rather than theoretical reasoning based on untestable assumptions.

E.2.3
Galileo also applied mathematics to accelerating objects, based on the distance covered in constant time intervals, and proved that falling objects have constant acceleration (and that acceleration under gravity was independent of mass).

E.2.4
At the time of Newton's birth, the heavens were considered separate from the earth, obeying different, unrelated laws. There was assumed to be no connection between the laws governing planes and objects on earth, and 'up there' no force was required for continual circular motion (clearly the planets kept circling with no force, as no one knew about gravity).

E.2.5
(This is just getting silly)

Newton's Pricipia Mathematica Philosopha Naturalis was written in Latin, very formal, with geometric derivations of laws, and corollaries. Contained definitions used in his laws, which connected and quantified the motions of all objects in the universe.

E.2.6
Newton defined the following:

Mass : Quantity of matter

Momentum : Quantity of motion

Impressed force : The force that induced a change in the quantity of motion (momentum).

Newton's laws were mainly formulated in terms of changes in momentum.

E.2.7
Newton's arguments for universal gravitation were based on the idea that planets were pulled along, and around their orbits by a centripetal force (this idea was partly taken from the work of Robert Hooke). Newton showed that an inverse square force could provide the force necessary for such motion, and coined the term centripetal. A central force is necessary for a object to move in a circle because a tangential force would just make the plant fly off in a straight line, rather than orbiting in a circle (or actually, an ellipse).

E.2.8
Under the law of universal gravitation, objects on the surface of the earth accelerate towards the center, as does the moon, though with the moon, the acceleration changes its direction rather than its speed though both accelerate towards the centre of the earth.

Since F = moa and F = Gmemo/r2, we can derive a = Gme/r2, thus by finding r and a for an object on the surface of earth, and r for the moon, a for the moon can be found.

E.2.9
Galileo's work was confined to describing the motion of objects near the earth's surface, and specifically that objects didn't require a continual force to keep moving (in a perfect, frictionless system). Newton developed a more general, and more mathematical basis for this idea.

E.2.10
The difference between having a law describing something and explaining the source of something: Newton could explain the behaviour of and predict gravity, but he could not explain its source (and we still can't really).

E.2.11
Newtonian mechanics implied (and Newton believed) that by knowing the starting position of every object in the universe, future states could be extrapolated. Newtonian mechanics is a deterministic, mechanical view of the universe.

E.2.12
Maths is important in physics in that it provides a modeling system which can describe natural phenomena, and make predictions form it.

E.3.1
Phlogistin/Caloric : This was a theory of heat from the mid 1800s. Phlogistin was a substance contained within things, which, when burnt, was removed as heat, thus turning them into ashes. Caloric was the original name for heat, which was thought to be like a fluid, flowing from hot objects to cold ones. This theory could not, however, explain the production of heat through friction.

E.3.2
The Newcombe engine was the first steam powered engine, and although being hopelessly inefficient, it effectively sparked the industrial age. Water was heated, turned into steam, which increased the pressure in the pipes above. This pressure was then used to push a piston down. On reaching the end, some steam was allowed to escape, and the piston was pushed back up by other pistons connected to the crank shaft. The steam which had escaped was run through and back to the water tank to be reused. The piston made use of several valves, which regulated the steam flow. Steam was only allowed to enter as the piston was moving down, and only allowed to leave as it was moving up.

E.3.3
The most major of Watt's modifications to this system was the use of a separate condenser. After escaping from the piston, the steam was kept under pressure and carried to a separate condenser. This meant that the boiler and piston could both be kept at a high pressure and similar temperature, and reduced the energy loss due to the latent heat of vaporisation. In addition to this, he developed systems to convert the back and forth motion of the pistons into rotary motion, through a system of gears. Another major development was the ability to regulate steam production, allowing the speed of the system to be regulated without reducing pressure (i.e. leaking steam).

E.3.4
Sadi Carnot developed a theoretical model of the heat exchange occurring to explain how heat moved through the system. He also developed the theoretical Carnot engine, which was 100% efficient, and so could be used to compare the efficiency of real engines.

E.3.5
Julius Mayer was the first to propose the idea that heat was actually a form of energy, which opposed the caloric theory. This idea was developed by Rumford and Joule, who worked to find the mechanical equivalent of heat.

E.3.6
Joule eventually produced a successful experiment which used a falling mass to turn paddles and produced a measurable change in temperature. The change in energy of the falling mass could be found, and this energy was in part being given to the water by the turning paddles. Based on this, Joule unified the two forms of energy, and is remembered in the unit of energy, joules.

E.3.7
Energy is an abstract concept, a way of viewing the system rather than an observable property which can be seen and analyzed, like mass or velocity. This, however, is useful in that is works at a more fundamental level and allows different forms of energy to be unified.

E.4.1
The wave theory fails to explain the photoelectric effect, because it is a continuous form of energy, and the photoelectric effect only makes sense if the energy from light is in quanta, and the amount of energy is determined by the frequency of the light.

E.4.2
Classical physics said that the energy in light was continuous, but this did not fit experimental observations. Plank attempted to modify the classical model by allowing for continuous incoming energy, but suggested that the output energy, the photo electrons, could only take certain amounts of energy. Einstein, however, created a completely new model, where both in incoming light radiation and the emission of photoelectrons were quantised (i.e., could only take certain values).

E.4.3
The de Broglie hypothesis was an extension of Einstein's linking of waves and quantization, by the suggestion that all particles have wave properties by the equation λ = h/p, or wavelength = plank's constant/momentum (thus it also equals h/mv).

E.4.4
de Broglie's ideas were eventually experimentally proved in relation to the diffraction of electrons, being particles with rest mass (they were diffracted through crystals with microscopic lattice spacing, and produced a diffraction pattern consistent with the above wavelength).

E.4.5
Schrodinger's wave theory analyzed the wave/particle duality, and attempted to combine it into one coherent model. One of the functions of this theory was to produce probability regions (in space-time) around atoms for where electrons would be (probably). This could then be extended to explain the diffraction pattern formed even by single electrons passing through a slit.

E.4.6
Schrodinger's wave theory was fundamentally an abstract model, which describes occurrences in a way that has no obvious link to the real world, and yet can effectively describe it. This de-coupling allows for modelling of events in such a way as to simply predict outcomes, rather than to represent the real world. Thus, it is easier to model the real world through abstract representations.