The Origin of the efractive Index 31-1 The index of refraction We have said before that light goes slower in water than in air, and slower, 31-1 The index of refraction slightly, in air than in vacuum. This effect is described by the index of refraction n. Now we would like to understand how such a slower velocity could come about 31-2 The field due to the material In particular, we should try to see what the relation is to some physical assumptions, 31-3 Dispersion or statements we made earlier, which were the following 31-4 Absorption (a) That the total electric field in any physical circumstance can always be epresented by the sum of the fields from all the charges in the universe 31-5 The energy carried by an electric wave b)That the field from a single charge is given by its acceleration evaluated with a retardation at the speed c, always(for the radiation field 31-6 Diffraction of light by a screen But, for a piece of glass, you might think: Oh, no, you should modify all this. You should say it is retarded at the speed c/n. That, however, is not right, and we have to understand why it is not It is approximately true that light or any electrical wave does appear to travel at the speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges--including the charges moving in he material-and with these basic contributions of the field travelling at the ultimate velocity c. Our problem is to understand how the apparently slower velocity comes about We shall try to understand the effect in a very simple case. A source which we shall call"the external source"is placed a large distance away from a thin plate of transparent material, say glass. We inquire about the field at a large distance on the opposite side of the plate. The situation is illustrated by the diagram of Fig. 31-1, where S and P are imagined to be very far away from the plate. According to the principles we have stated earlier, an electric field anywhere n5 that is far from all moving charges is the(vector) sum of the fields produced by the external source (at S) and the fields produced by each of the charges in the plate c of glass, every one with its proper retardation at the velocity c. Remember that the contribution of each charge is not changed by the presence of the other charges These are our basic principles. The field at P can be written thus Fig. 31-1. Electric waves passing through a layer of transparent materia y Eeach charge (311) E=E+∑ Eeach charges here es is the field due to the source alone and would be precisely the field at P if there were no material present. We expect the field at P to be different from Eg if there are any other moving charges Why should there be charges moving in the glass? We know that all material consists of atoms which contain electrons. When the electric field of the source acts on these atoms it drives the electrons up and down, because it exerts a force on the electrons. And moving electrons generate a field-they constitute new radiators These new radiators are related to the source S, because they are driven by the field of the source. The total field is not just the field of the source S, but it is modified by the additional contribution from the other moving charges. This means that the field is not the same as the one which was there before the glass as there but is modified, and it turns out that it is modified in such a way that 31-1
the field inside the glass appears to be moving at a different speed That is the idea which we would like to work out quantitatively Now this is, in the exact case, pretty complicated, because although we have id that all the other moving charges are driven by the source field, that is not quite true. If we think of a particular charge, it feels not only the source, but like anything else in the world, it feels all of the charges that are moving. It feels, in particular, the charges that are moving somewhere else in the glass. So the total field which is acting on a particular charge is a combination of the fields from th can see that it would take a complicated set of equations to get l es doing!You complete and exact formula. It is so complicated that we postpone this problem until next year. Instead we shall work out a very simple case in order to understand all the physical principles very clearly. We take a circumstance in which the effects from the other atoms is very small relative to the effects from the source. In other words e take a material in which the total field is not modified very much by the motion of the other charges. That corresponds to a material in which the index of refraction is very close to 1, which will happen, for example, if the density of the atoms is very low. Our calculation will be valid for any case in which the index is for any reason very close to 1. In this way we shall avoid the complications of the most general, complete solution Incidentally, you should notice that there is another effect caused by the motion of the charges in the plate. These charges will also radiate waves back toward the source S. This backward-going field is the light we see reflected from the surfaces transparent materials. It does ne t the surface The backw radiation comes from everywhere in the interior, but it turns out that the total effect is equivalent to a reflection from the surfaces. These reflection effects are beyond our approximation at the moment because we shall be limited to a calculation for a material with an index so close to l that very little light is reflected Before we proceed with our study of how the index of refraction comes about we should understand that all that is required to understand refraction is to under stand why the apparent wave velocity is different in different materials. The bending of light rays comes about just because the effective speed of the waves is different in the materials. To remind you how that comes about we have drawn in Fig. 31-2 several successive crests of an electric wave which arrives fre vacuum onto the surface of a block of glass. The arrow perpendicular to the wave crests indicates the direction of travel of the wave. Now all oscillations in the wave must have the same frequency. (We have seen that driven oscillations have the same frequency as the driving source. ) This means, also, that the wave crests for the waves on both sides of the surface must have the same spacing along the surface because they must travel together, so that a charge sitting at the boundary will Fig. 31-2. Relation between refrac- feel only one frequency. The shortest distance between crests of the wave, however, on and velocity change is the wavelength which is the velocity divided by the frequency. On the vacuum side it is o=2Tc/w, and on the other side it is x= 2y/w or 2c/ is the velocity of the wave. From the figure we can see that the only way for the vaves to"fit"properly at the boundary is for the waves in the material to be travelling at a different angle with respect to the surface. From the geometry of the figure you can see that for a"fit"we must have Ao/sin 00=A/sin B, or sin 0o/sin 0= n, which is Snell's law. We shall, for the rest of our discussion consider only why light has an effective speed of c/n in material of index n, and the bending of the We go back now to the situation shown in Fig. 31-1. We see that what we have to do is to calculate the field produced at P by all the oscillating charges in the glass plate. We shall call this part of the field Ea, and it is just the sum written as the second term in Eq (31.2). When we add it to the term Es, due to the source, we will have the total field at P
This is probably the most complicated thing that we are going to do this year, but it is complicated only in that there are many pieces that have to be put to- gether; each piece, however, is very simple. Unlike other derivations where we say, "Forget the derivation, just look at the answer!, "in this case we do not need the answer so much as the derivation In other words, the thing to under stand now is the physical machinery for the production of the index To see where we are going, let us first find out what the"correction field" would have to be if the total field at P is going to look like radiation from the source that is slowed down while passing through the thin plate. If the plate had no effect on it, the field of a wave travelling to the right(along the z-axis) would be E。= Eo cos a(t-z/c) or, using the exponential notation (314 Now what would happen if the wave travelled more slowly in going through the plate? Let us call the thickness of the plate Az. If the plate were not there the rave would travel the distance Az in the time Az/c. But if it appears to travel at the speed c/n then it should take the longer time n Az/c or the additional time At=(n-1)4z/c. After that it would continue to travel at the speed c again We can take into account the extra delay in getting through the plate by replacing t in Eq (31.4)by(t-)or by [! -(n-1)Az/c]. So the wave after insertion of the plate should be written We can also write this equation as Enftor plate e -ia(n-1)az/e Eoe(/e), which says that the wave after the plate is obtained from the wave which could exist without the plate, i, e, from Es, by multiplying by the factor e-i(n-m)42/e Now we know that multiplying an oscillating function like e t by a factor e just says that we change the phase of the oscillation by the angle which is, of course what the extra delay in passing through the thickness 4z has done. It has retarded the phase by the amount w(n- D)Az/c(retarded, because of the minus sign the exponent). e have said earlier that the plate should add a field ea to the original field Es eoe but we have found instead that the effect of the plate is to multiply the field by a factor which shifts its phase. However, that is really all right because we can get the same result by adding a suitable complex number. It is articularly easy to find the right number to add in the case that Az is small, for you will remember that if x is a small number then e is nearly equal to(1 x We can write, therefore Using this equality in Eq.(31.6), we have Eafter plate= eoe ie(t-a/c)_iw(n 1)△z Ee(-21).(31.8) En The first term is just the field from the source, and the second term must just be Rool Axis qual to Ea, the field produced to the right of the plate by the oscillating charges of the plate--expressed here in terms of the index of refraction n, and depending, What we have been doing is easily visualized if we look at the complex number i Fig. 31-3. Diagram for the trans- of course, on the strength of the wave from the source ed wave at a particular t and z. diagram in Fig. 31-3. We first draw the number es (we chose some values for z and t so that es comes out horizontal, but this is not necessary). The delay due to
lowing down in the plate would delay the phase of this number, that is, it would rotate Es through a negative angle. But this is equivalent to adding the small vector ea at roughly right angles to Eg. But that is just what the factor -i means in the second term of Eq.(31.8). It says that if Es is real, then Ea is negative imaginary or that, in general, Es and Ea make a right angle. 31-2 The field due to the material We now have to ask: Is the field ea obtained in the second term of Eq. (31.8) the kind we would expect from oscillating charges in the plate? If we can show that it is, we will then have calculated what the index n should be! [Since n is the only nonfundamental number in Eq(31.8). We turn now to calculating what field Ea the charges in th (To help keep track of many symbols we have used up to now, and will be using in the rest of our calcula tion, we have put them all together in Table 31-1.) Symbols used in the calculations Ea= field produced by charges in the plate plate z= perpendicular distance from the plate index of refraction w= frequency(angular)of the radiation charges per unit volume in the pla m= number of charges per unit area of the plate ge charge on an electron wo- resonant frequency of an electron bound in an atom If the source S(of Fig 31-1)is far off to the left then the field Es will hay the same phase everywhere on the plate, so we can write that in the neighborhood f the plate Es= Ec (31.9) Right at the plate, where z=0, we will have Es (at the plate) Each of the electrons in the atoms of the plate will feel this electric field and will be driven up and down (we assume the direction of Eo is vertical) by the electric force ge. To find what motion we expect for the electrons, we will assume that the atoms are little oscillators, that is, that the electrons are fastened elastically to the atoms, which means that if a force is applied to an electron its displacement from its normal position will be proportional to the force. You may think that this is a funny model of an atom if you have heard about electrons whirling around in orbits. But that is just an oversimplified picture ven by the theory of says that, so far as problems involving light are concerned, the electrons behave as though they were held by springs. So we shall suppose that the electrons have a linear restoring force which, together with their mass m, makes them behave like little oscillators, with a resonant frequency wo. We have already studied such cillators, and we know that the equation of their motion is written this way here F is the driving for
For our problem, the driving force comes from the electric field of the wave from the source, so we should us f= gees- eee, where qe is the electric charge on the electron and for Es we use the expression Es= eoew from(31. 10). Our equation of motion for the electron is then geese We have solved this equation before, and we know that the solution is (31.14) where, by substituting in(31. 13), we find that geO (31.15) geO (31.16) We have what we needed to know-the motion of the electrons in the plate. And it is the same for every electron, except that the mean position(the "zero"of the motion)is, of course, different for each electron Now we are ready to find the field ea that these atoms produce at the point P, because we have already worked out(at the end of Chapter 30)what field is pro- duced by a sheet of charges that all move together. Referring back to Eq. (30.19), we see that the field ea at P is just a negative constant times the velocity of the charges retarded in time the amount z/c. Differentiating x in Eq. ( 31. 16) to get the velocity, and sticking in the retardation [or just putting xo from (31.15) into(30. 18)] yield eoc ia qeEo wo(4-2/e) (31.17 Just as we expected, the driven motion of the electrons produced an extra wave which travels to the right(that is what the factor eiw -z/e)says), and the amplitude of this wave is proportional to the number of atoms per unit area in the plate (the factor n)and also proportional to the strength of the source field(the factor Eo). Then there are some factors which depend on the atomic properties (qe, m, and wo), as we should expect n. The two expressions will, in fact, be identical i rial wha,o The most important thing, however, is that this formula(31. 17) for Ea looks very much like the expression for ea that we got in Eq. (31.8)by saying that the original wave was delayed in passing through a mat (n-1)△z=-m (31.18) Notice that both sides are proportional to Az, since n, which is the number of atoms per unit area, is equal to N AZ, where N is the number of atoms per unit volume of the plate. Substituting N AZ for n and cancelling the Az, we get our main ul, a formula for the index of refraction in terms of the properties of the atoms of the material-and of the frequency of the light 1+ (31.19) This equation gives the"explanation "of the index of refraction that we wished to obt 31-5
31-3 Dispersion Notice that in the above process we have obtained ng ve For we have not only a number for the index of refraction which can be computed F from the basic atomic quantities, but we have also learned how the index of refraction should vary with the frequency w of the light. This is something we would never understand from the simple statement that " light travels slower in a transparent material. "We still have the problem, of course, of knowing how many atoms per unit volume there are, and what is their natural frequency wo. We do not know this just yet, because it is different for every different material, and we cannot get a general theory of that now. Formulation of a general theory of the properties of different substances-their natural frequencies, and so on-i possible only with quantum atomic mechanics. Also, different materials have different properties and different indexes, so we cannot expect, anyway, to get a general formula for the index which will apply to all substances However, we shall discuss the formula we have obtained, in various possible circumstances. First of all, for most ordinary gases(for instance, for air, most colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron oscillators correspond to ultraviolet light. These frequencies are higher than the frequencies of visible light, that is, wo is much larger than w of visible light, and to a first approximation, we can disregard w 2 in comparison with a3. Then we find that the index is nearly constant. So for a gas, the index is nearly constant. Thi is also true for most other transparent substances, like glass. If we look at our expression a little more closely, however, we notice that as w rises, taking a little bit more away from the denominator, the index also rises. So n rises slowly with frequency. The index is higher for blue light than for red light. That is the reason why a prism bends the light more in the blue than in the red The phenomenon that the index depends upon the frequency is called the phenomenon of dispersion, because it is the basis of the fact that light is "dispersed by a prism into a spectrum. The equation for the index of refraction as a function of frequency is called a dispersion equation. So we have obtained a dispersion equa- tion.(In the past few years"dispersion equations"have been finding a new use in the theory of elementary particles. Our dispersion equation suggests other interesting effects. If we have a natural frequency wo which lies in the visible region, or if we measure the index of refraction of a material like glass in the ultraviolet, where w gets near wo, we see that at frequencies very close to the natural frequency the index can get enor mously large, because the denominator can go to zero. Next, suppose that w is greater than wo. This would occur, for example, if we take a material li say,and shine x-ray radiation on it. In fact, since many materials which are to visible light, like graphite for instance, are transparent to X-rays, we can also Ik about the index of refraction of carbon for All the natural fre of the carbon atoms would be much lower than the frequency we are using in the X-rays, since x-ray radiation has a very high frequency. The index of refraction is that given by our dispersion equation if we set wo equal to zero(we neglect wl comparison with A similar situation would occur if we beam radiowaves(or light) ree electrons. In the upper atmosphere electrons are liberated from their atoms by ultraviolet light from the sun and they sit up there as free electrons. For free electrons wo=o(there is no elastic restoring force). Setting sion equation yields the correct formula for the index of refraction for radiowaves in the stratosphere, where N is now to represent the density of free electrons(num r per unit volume)in the stratosphere. But let us look again at the equation, if we beam x-rays on matter, or radiowaves(or any electric waves)on free electrons ) becomes negative, and we obtain the result that n is less than one. That means that the effective speed of the waves in the substance is faster than c! Can that be c It is correct. In spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than 1. This
just means that the phase shift which is produced by the scattered light can be either positive or negative. It can be shown, however, that the speed at which you an send a signal is not determined by the index at one frequency, but depends on what the index is at many frequencies. What the index tells us is the speed at which he nodes(or crests)of the wave travel. The node of a wave is not a signal by itself. In a perfect wave, which has no modulations of any kind, i. e,, which is a steady oscillation, you cannot really say when it"starts, " so you cannot use it for a timing signal. In order to send a signal you have to change the wave somehow, make a notch in it, make it a little bit fatter or thinner. That means that you have to have more than one frequency in the wave, and it can be shown that the speed at which signals travel is not dependent upon the index alone, but upon the way that the index changes with the frequency. This subject we must also delay (until Chapter 48). Then we will calculate for you the actual speed of signals through such a piece of glass, and you will see that it will not be faster than the speed of light, although the nodes, which are mathematical points, do travel faster than the speed of light. Just to give a slight hint as to how that happens, you will not rea difficulty has to do with the fact that the responses of the charges are opposite to the field, i.e., the sign has gotten reversed. Thus in our expression for x(eq. 31. 16) the displacement of the charge is in the direction opposite to the driving field, because(wb -w2)is negative for small wo. The formula says that when the electric field is pulling in one direction, the charge is moving in the opposite direc How does the charge happen to be going in the opposite direction? It certainly does not start off in the opposite direction when the field is first turned on.When the motion first starts there is a transient, which settles down after awhile, a only then is the phase of the oscillation of the charge opposite to the driving field And it is then that the phase of the transmitted field can appear to be advanced with respect to the source wave. It is this advance in phase which is meant when we say that the"phase velocity "or velocity of the nodes is greater than c. In Fig 31-4 we give a schematic idea of how the waves might look for a case where the wave is suddenly turned on(to make a signal). You will see from the diagram that the signal (i. e. the start of the wave)is not earlier for the wave which ends up with an advance in phase Fig. 31-4. Wave"signals deloy of phose Let us now look again at our dispersion equation. We should remark that our analysis of the refractive index gives a result that is somewhat simpler than you would actually find in nature. To be completely accurate we must add some refinements. First, we should expect that our model of the atomic oscillator should have some damping force(otherwise once started it would oscillate forever, and we do not expect that to happen). We have worked out before(Eq. 23. 8)the motion of a damped oscillator and the result is that the denominator in Eq (31.16), and therefore in (31. 19), is changed from(ab3-w2)to(ao-w2+ ira), where n is the damping coefficient. We need a second modification to take into account the fact that there are several resonant frequencies for a particular kind of atom. It is easy to fix up
dispersion equation by imagining that there are several different kinds of oscil- lators, but that each oscillator acts separately, and so we simply add the contri- butions of all the oscillators. Let us say that there are Nk electrons per unit of lume whose natural frequency is wk and whose damping factor is tk. We would then have for our dispersion equation n=1 N w"+iRa We have, finally, a complete expression which describes the index of refraction that is observed for many substances. The index described by this formula varies with frequency roughly like the curve shown in Fig.31-5 You will note that so long as w is not too close to one of the resonant frequen- cies, the slope of the curve is positive. Such a positive slope is called"normal" ec dispersion(because it is clearly the most common occurrence). Very near the resonant frequencies, however, there is a small range of o' s for which the slope is negative. Such a negative slope is often referred to as"anomalous"(meaning abnormal) dispersion, because it seemed unusual when it was first observed, long Fig. 31-5. The index of refraction as before anyone even knew there were such things as electrons. From our point of a function of frequency. view both slopes are quite"normal""! 31-4 Absorption Perhaps you have noticed something a little strange about the last form Eq. 31. 20)we obtained for our dispersion equation. Because of the term i we put in to take account of damping, the index of refraction is now a comple number!What does that mean? By working out what the real and imaginary parts of n are we could write (31.21) where n'and n"are real numbers. (We use the minus sign in front of the in because then n"will turn out to be a positive number, as you can show for yourself.) We can see what such a complex index means by going back to Eq. (31.6) which is the equation of the wave after it goes through a plate of material with an index n. If we put our complex n into this equation, and do some rearranging, we n"△x/e。-i(n'-1)△zlep,i(t-2/c) (31.22) The last factors, marked B in Eq.(31. 22), are just the form we had before, and again describe a wave whose phase has been delayed by the angle w(n'-1)4z/c in traversing the material. The first term(A)is new and is an exponential factor with a real exponent, because there were two is that cancelled. Also, the exponent is negative, so the factor is a real number less than one. It describes a decrease in the magnitude of the field and, as we should expect by an amount which is more the larger 4z is. As the wave goes through the material, it is weakened. The material is ""absorbing"part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy. We see that the imaginary part n"of a complex index of refraction represents an absorption(or"attenuation")of the wave. In fact, n"is sometimes eferred to as the" absorption index We may also point out that an imaginary part to the index n corresponds to bending the arrow Ea in Fig. 31-3 toward the origin. It is clear why the transmitted field is then decreased Actually, although in quantum mechanics Eg. (34.20)is still valid hydrogen, has several resonant frequencies. Therefore N is not really the number of electrons having the frequency wk, but is replaced instead by Nfk, where N is the number of atoms per unit volume and fk(called the oscillator strengthis a factor that tells how strongly the atom exhibits each of its resonant fr
Normally, for instance as in glass, the absorption of light is very small This is to be expected from our Eq. (31.20), because the imaginary part of the denominator, iGg is much smaller than the term(wk-w2). But if the light fre- quency w is very close to wk then the resonance term (wk-w2)can become small compared with inw and the index becomes almost completely imaginary. The bsorption of the light becomes the dominant effect. It is just this effect that gives the dark lines in the spectrum of light which we receive from the sun. The light from the solar surface has passed through the suns atmosphere (as well as the earths), and the light has been strongly absorbed at the resonant frequencies of the atoms in the solar atmosphere The observation of such spectral lines in the sunlight allows us to tell the resonant frequencies of the atoms and hence the chemical composition of the sun's atmosphere. The same kind of observations tell us about the materials in the stars From such measurements we know that the chemical elements in the sun and in the stars are the same as those we find on the earth 31-5 The energy carried by an electric wave shal e have seen that the imaginary part of the index means absorption.We now use this knowledge to find out how much energy is carried by a light wave. We have given earlier an argument that the energy carried by light is proportional to E2, the time average of the square of the electric field in the wave The decrease in E due to absorption must mean a loss of energy, which would go into some friction of the electrons and ht guess, would end up as heat in the materia If we consider the light arriving on a unit area, say one square centimeter, of our plate in Fig. 31-1, then we can write the following energy equation(if we assume that energy is conserved, as we do Energy in per sec energy out per sec work done per sec. (31.23) For the first term we can write aEi, where a is the as yet unknown constant of proportionality which relates the average value of E- to the energy being carried For the second term we must include the part from the radiating atoms of the material, so we should use a(eg ea)2, or(evaluating the square)a(E? 2E,Ea E2) All of our calculations have been made for a thin layer of material whose index is not too far from 1, so that ea would always be much less than Es just to make the calculations easier). In keeping with our approximations, we should, therefore, leave out the term E because it is much smaller than eea. You may say: "Then you should leave out E, Ea also, because it is much smaller than E? It is true that E, ea is much smaller than E3, but we must keep e, ea or our appro mation will be the one that would apply if we neglected the presence of the material ecking that our calculations are consistent is to see that we always keep terms which are proportional to N Az, the area density of atoms in the material, but we leave out terms which are proportional to(N 4z)2 or any higher power of N Az. Ours is what should be called a"low-density approxi nation In the same spirit, we might remark that our energy equation has neglected the energy in the reflected wave. But that is OK because this term, too, is propor tional to(N Az),, since the amplitude of the reflected wave is proportional to N△z For the last term in Eq. (31. 23)we wish to compute the rate at incoming wave is doing work on the electrons. We know that work is force times distance, so the rate of doing work(also called power )is the force times the veloc ity. It is really F.V, but we do not need to worry about the dot product when the velocity and force are along the same direction as they are here(except for a possible sign). So for each atom we take ge E,u for the average rate of
oing work. Since there are N Az atoms in a unit area, the last term in Eq.(31.23) aE=aE?+ 2aEs Ea t Naz qe Esu (3124 The E terms cancel, and we have 2aEE=N△zqE2 We now go back to Eq (30.19), which tells us that for large z ea N△zqe (ret by z/c) ( recalling that n=N Az). Putting Eq.(3126)into the left-hand side of (31 25) NAzqe e(at z)v(ret by z/c) However, Es(at z)is Es(at atoms)retarded by z/c. Since the average pendent of time, it is the same now as retarded by z/c, or is Es(at atom).v, the same average that appears on the right-hand side of (31. 25). The two sides are efore equal if (3127) We have discovered that if energy is to be conserved, the energy carried in an elec- tric wave per unit area and per unit time (or what we have called the intensity) must be given by EoCEZ. If we call the intensity S,we have CE2 nergy/area/time opaque screen where the bar means the time average. We have a nice bonus result from our of the refractive index! S EEs E=es 31-6 Diffraction of light by a screen It is now a good time to take up a somewhat different matter which handle with the machinery of this chapter. In the last chapter we said that when c you have an opaque screen and the light can come through some holes, the distribu tion of intensity-the diffraction pattern- could be obtained by imagining instead that the holes are replaced by so (oscillators)uniformly distributed over the wofl+Eplug"o hole. In other words, the diffracted wave is the same as though the hole were a new source. We have to explain the reason for that, because the hole is, of course just where there are no sources, where there are no accelerating charges Diffraction by a screen Let us first ask: "What is an opaque screen? "Suppose we have a completely paque screen between a source S and an observer at P, as in Fig. 31-6(a).If the creen is"opaque" there is no field at P. Why is there no field there? According to the basic principles we should obtain the field at P as the field es of the source delayed, plus the field from all the other charges around. But, as we have seen above, the charges in the screen will be set in motion by the field Ea, and these motions generate a new field which, if the screen is opaque, must exactly cancel the field Es on the back side of the screen. You say: What a miracle that it bal ances exactly! Suppose it was not exactly right! " If it were not exactly right(re- member that this opaque scree has some thickness ), the field toward the part of the screen would not be exactly zero. So, not being zero, it would set into motion some other charges in the material of the screen, and thus make a little more field trying to get the total balanced out. So if we make the screen thick enough, there is no residual field, because there is enough opportunity to finally get the thing quieted down. In terms of our formulas above we would say that the
