EM Waves in Media. What happens when an EM wave travels through our model material?

Transcription

1 EM Waves in Media We can model a material as made of atoms which have a charged electron bound to a nucleus by a spring. We model the nuclei as being fixed to a grid (because they are heavy, they don t move much). What happens when an EM wave travels through our model material? Griffiths works this model out in detail (and you ll see it in 401 or 454). We won t try to actually solve the equations today, we ll just talk about the solutions.

2 EM Waves in Media (2) The electric field of the wave applies a force to the electrons because they are charged, and makes them move. The nuclei are also charged, but don t move There is an equilibrium point, where the electric field force is equal to the spring-force (which grows with displacement). There is a natural frequency of motion of the electron mass on the spring:ω = k m. If the frequency of the EM wave is low compared to the mass-spring frequency, the electron basically sits at the equilbrium point. It moves back and forth at the the frequency of the wave as the equilibrium point moves. The motion of the charges is a current, so it produces a magnetic field. The current produces a magnetic field. This time-dependent magnetic field makes an electric field. So we have find self-consistent wave solutions of Maxwell s Equations in the presence of the movable charges.

3 EM Waves in Media (3) When you work out the details, the medium behaves like it has a dielectric constant ε greater than vacuum ε 0. EM waves travel slower, and the relation between the magnitudes of E and B is altered. If there is damping of the electrons, the motion dissipates more energy. The only place the energy can come from is the wave fields. So the wave is attenuated. The material is not transparent. As the frequency approaches the resonant frequency of the electrons, the motion of the charges gets bigger. The bigger motion is a bigger current, which changes the wave solutions more. The wave slows down farther, and the absorption gets larger. When the frequency is much higher than the resonant frequency, the mass of the charges keeps them from moving very much. They just don t have time to move far before the field reverses. So at higher frequency, the medium acts more like vacuum again.

4 Index of Refraction, and Dispersion For a material to be transparent to light, the resonant frequencies have to be higher than optical frequencies. This model predicts that shorter wavelengths have slower propagation velocity because they are closer to the resonances above optical frequencies. The index of refraction of a material c/v. So the index of refraction goes up as the velocity goes down at higher frequencies. This is exactly what happens in glass, water, etc. Blue light has a higher frequency than red light, so it goes slower, thus is refracted more. This is how prisms work. Because different colors are bent differently, or dispersed, the effect of frequency-dependent velocity is called dispersion.

5 Electron in a Magnetic Field If there is a magnetic field present, there is a magnetic force as well as a spring force on the electrons. The magnetic force is r F = q r v r B If the wave is travelling in the Z direction, the electric field is in the X-Y plane, and the velocity of the electtrons will be in the XY plane. If magnetic field is in the Z direction, the electron velocity is a right angles to it, so the cross product will be non-zero. The case that will interest us is if the motion is initially circular in the x-y plane. The magnetic force is at right angles to the motion, so it s either toward or away from the center of the circle

6 Interaction of Circularly Polarized Light with Media in Magnetic Fields Circularly polarized light makes the charges orbit in one direction for right-handed polarization and the other for lefthanded polarization. If we apply a static magnetic field parallel to the light propagation to our oscillators, we have different effective spring constants for left and right circular motions. So we have different effects of the material on the light wave for left and right circular polarization. In particular, the wave velocity is different for left and right circular.

7 Faraday Effect The motion of electrons in the medium causes the speed of light to be slower than it is in vacuum. If we have linearly polarized light entering the sample, we can think of that as a sum of left and right circular polarized light, that are in phase with each other. Circularly polarized light has an electric field that rotates in time. This makes electrons try to move in circles. The magnetic field affects the motion differently for electrons circling to the left and the right. So the speed of light is different for left- and right-circular polarization when there is a magnetic field. If the velocities of left- and right-circular polarized light are different, they will be out of phase when they leave the sample. When we add them back up again at the end of the sample, the relative phase difference between left and right circular results in a linear polarization at a rotated angle. This is the Faraday Effect.

8 Error Propagation It is seldom possible to measure an interesting physical quantity directly. Usually we have to make several measurements and combine them in some way. The measurements all have uncertainties (errors), and they all contribute to the uncertainty in the final computed result. But how do we compute the error on the final result? The definition of the mean and standard deviation are m = 1 σ 2 [ m] = 1 m i i=1 i=1 ( m i m ) 2 ow, instead of a direct measurement, consider a function f ( x i, y i ) of two measurements x and y. Plugging this into the formulas, we get σ 2 f ( x, y) = 1 [ f ( x, y) ] = 1 i=1 i=1 f ( x i, y i ) ( ( ) ) 2 f ( x i, y i ) f x, y

9 Error Propagation (2) Let s Taylor expand the function around x and y, the mean values of the two measurements f ( x i, y i ) f ( x, y ) + f ( x x i x )+ f ( y y i y ) +... Plug this into the formula for the mean, and we get f ( x, y) = f ( x, y ) Plug this in to the standard deviation formula, and we get σ 2 [ f ( x, y) ] = 1 = 1 i=1 i=1 f ( x x i x ) + f ( y y i y ) f x +2 f x 2 ( x i x ) 2 + f y 2 f y x i x ( ) y i y ( ) 2 ( y i y ) 2

10 Error Propagation (3) Break the sum into 3 sums, and factor out the derivatives σ 2 [ f ( x, y) ] = f x + f y +2 f x 2 2 f y i=1 i=1 ( x i x ) 2 ( y i y ) 2 i=1 ( ) ( x i x ) y i y The first two sum brackets are just the standard deviations of x and y. If x and y are uncorrelated, which is usually the case if they are different types of measurments of different types of quantities, then on the average So the result is σ 2 [( x i x )( y i y )] = 0 i=1 [ f ] = f x 2 σ 2 [ x] + f y 2 σ 2 [ y] Of course we have to take the square root to get the error (standard deviation) instead of the variance.

11 Systematic Errors Systematic errors are the ones that don t get smaller just because you average more data together. There are two parts of any systematic error contribution the uncertainty of some input number the sensitivity of your result to that input number The sensitivity part is usually non-controversial meaning that anyone will agree how much your final answer will change if you change the input number. The input uncertainty is often controversial meaning that you have to make an educated guess about how sure you are about something. In the Faraday lab, the gaussmeter is claimed to have a 2% accuracy by the manufacturer. This normally means that there is a scale-factor systematic error. Presumably every magnetic measurement that you make could be 2% high. Or perhaps every measurement that you make is 2% low. This kind of statement usually doesn t mean that if you do repeated measurements of the same thing, changing absolutely nothing, that the results vary by 2%. You can check this: put the gaussmeter into the magnet on a support block so it doesn t move, then just watch the number to see if it varies. It probably won t.

12 Combining Errors Errors from different sources are normally uncorrelated: they have random signs. So it s overly pessimistic to just add the numbers together as if they all have the same sign. The generally accepted procedure is to add the errors in quadrature: square root of sum of squares σ total = σ σ σ σ The smaller error terms add surprisingly little: = 1.25 = = 1.01 = A second error that is half as big as the first adds only 12% to the total error, and a second error that is 10% as big as the first adds only half a percent! Sometimes the statistical error is insignificant because you can average enough data. If it is not insignificant, it is often quoted separately: X ± σ statistical ± σ systematic There is often a table of systematic error contributions in a real physics paper that break it down further.

13 Combining Measurements You should have 4 line-fits for the c parameter vs current or magnetic field, with 4 different slopes, and 4 different errors. You need to combine them into one slope and one error. The first thing to do is to plot the values and errors. With Gnuplot, make a data file with just 1,2,3,4,... in the first column, and your values and errors in the other columns. Hopefully, it looks something like this: 'test.dat' u 1:2: The values are clearly pretty consistent with each other within the errors.

14 Combining Consistent Measurements You should then perform a weighted average: x = w i x i w i This formula combines your data into a single number, regardless of what values you use for the weights. You can use standard error-propagation formulas to calculate the error on the weighted average. What weights should you use? If some of your measurements have much larger errors that the others, you probably don t want to give them as much weight as the better measurements. The weights that give the minimum error on the average have a simple relation to the measurement errors: w i = 1 σ i 2 The error on the average turns out to be σ[ x ] = 1 w = 1 2 i 1 σ i

15 Combining Consistent Measurements (2) The error on the weighted average is smaller than any of the individual errors. If the individual errors are all equal (or nearly equal), then the combined error turns out to be σ[ x ] = σ i This is a good sanity check if you are doing the calculation by hand. There is a trick to make Gnuplot do this calculation for you. Define a function with one parameter: f(x)=avg Then fit your data file (with a dummy first column, and including the errorbars) fit f(x) my.dat using 1:2:3 via avg The result of the fit is precisely the weighted average from the above formula.

16 Combining Consistent Measurements (3) But beware, the Gnuplot error isn t exactly the value calculated from the above formula. Gnuplot multiplies the right error by the RMS of the residuals (square root of chisquare per degree of freedom). This number is in the fit results, so you can divide it out to get the right error. You can draw the average on the plot you made above by replot avg You might also add another line to your data file, containing the average and its error: 'test.dat' u 1:2:3 avg

17 Bad Measurements It may happen that one of your measurements has a much larger error than the others: 'test2.dat' u 1:2: The bad measurement is actually only slightly more than one sigma away from the others, so it s not wrong given its error. It s OK to include it in the weighted average. Of course, the weight will be very small, so it doesn t matter much if you include it or not. Beware that in such a case, you should still look carefully to see if the good measurements are really consistent with each other. The autoscaling makes that hard to see.

18 Bad Measurements (2) The more troubling case is when one or more measurements are inconsistent given their errors: 'test2.dat' u 1:2: It is frowned upon to simply throw away data that doesn t fit your expectations, since this introduces biases. The first thing to do is check if there is some mistake in the bad point, like a typo, wrong file, failed fit,... Technically, you should make the same checks on all the data points, not just the one you don t like! If it s practical, you might repeat the experiment to see if the bad point reproduces. (obel Prizes have been won by finding out that the bad point was really right!)

19 Bad Measurements (3) If most of your measurements are inconsistent within their errors, you should check whether the errors you assigned are calculated correctly. If you are sure that your error bar calculations are correct, and you have really good reason to believe that you were really measuring the same thing each time and should have gotten the same result, you should probably ignore the error bars. But you still need to combine the data, and assign an error to the combined value. In this case, the sensible thing to do is just a straight average, and the error on the average is the usual: RMS of the distribution divided by square root of. ote that you can use Gnuplot to do this using nearly the same trick The only difference is that you fit the constant function ignoring the errors. The fit result will be the unweighted average, and the error on the unweighted average. The RMS of residuals will be the RMS of the distribution.

20 Bad Measurements (4) The most controversial case is when you have a handful of measurements with rather different errors, and one or two outliers that you can t explain away. You probably shouldn t use the weighted-average formula umodified, because it will give an error on the average that is smaller than any of the individual errors, even though there is a gross inconsistency in the data. But you don t have enough measurements done in a consistent way to use statistical techniques on the distribution of measurements. But you still need to do something! And it s possible to be in the gray area where the handful of measurements is neither consistent nor grossly inconsistent. In this case, the most important thing is to carefully explain what you have chosen to do, so someone using your result can judge whether to trust your result and error for his purposes or not.