CHAPTER 32: ELECTROMAGNETIC WAVES For those of you who are interested, below are the differential, or point, form of the four Maxwell s equations we studied this semester. The version of Maxwell s equations that we studied are known as the integral form. They involve vector differentiation (the first two use a divergence and the second two use a curl). Maxwell s original presentation (1861) of these equations involved 20 equations instead of 4. The vector calculus notation that you see below was developed by Heaviside and Gibbs around the late 1800s and the original 20 equations were reduced to these four by Heaviside around 1885. Faraday s law says that a time-varying magnetic field acts as a source of an electric field, and Ampere s law says that a time-varying electric field acts as a source of a magnetic field. These two phenomena can actually sustain one another, creating an electromagnetic wave that propagates thru space. Much of what we learned in Chapter 15 applies to electromagnetic waves, even though, physically, they are very different from mechanical waves. One huge difference is that electromagnetic waves do not require a material medium in order to propagate thru space this idea was so foreign to physicists at the time that many postulated the existence of a luminiferous aether, an invisible medium which visible light and other electromagnetic waves needed in order to be able to propagate thru space. Maxwell s equations predict that stationary charges produce electrostatic fields. They also predict that charges moving at constant velocity produce
magnetostatic fields. These static fields can be analyzed independent of one another. We need for one of these fields to vary in time (not static) in order to induce the other, so that these fields are coupled meaning the behavior of one field influences that of the other. It turns out that, according to Maxwell s equations, an accelerating charge will produce a time-dependent, self-sustaining electromagnetic field that propagates thru space as a transverse wave. The electric and magnetic fields oscillate perpendicular to the direction the wave is traveling, and the amplitudes of these oscillations are the maximum values of the fields themselves. Thus, electromagnetic waves are not completely divorced from matter you do need an electric charge in order to generate an electromagnetic wave, but sustaining this type of wave does not require the acceleration of any adjacent charges. It propagates thru space whether or not there is any matter present other than the accelerating charge which produced it. As with mechanical waves, one way to generate electromagnetic waves (I will abbreviate this as EM waves for the remainder of the lecture) is thru simple harmonic motion, which means the displacement of the source from equilibrium oscillates sinusoidally (in time) at a well-defined frequency. The term electromagnetic radiation is used interchangeably with electromagnetic waves since the fields radiate away from the source (some accelerating distribution of charge). Maxwell s equations also predict a wave equation for both the electric and the magnetic fields where the speed c (see the discussion concerning the wave equation in the CH 15 notes) is given by 1 c 2 = ε 0μ 0. As with mechanical waves, the constant wavespeed c obeys the relationship c = λf. For EM waves propagating thru empty space (vacuum), c = 3.00 x 10 8 m/s. EM waves span a wide range of frequencies (and, thus, a wide range of corresponding wavelengths) and what we can visually perceive is called visible light and this occupies a very narrow band of the electromagnetic spectrum shown in the figure below. Note the wavelengths/frequencies are depicted using a logarithmic scale (as opposed to a linear scale).
Plane Electromagnetic Waves A plane wave is one whose properties do not vary in directions perpendicular to the direction that the wave travels For plane EM waves this means that, at any instant in time, on any plane perpendicular to the EM wave s direction of propagation, the electric and magnetic fields are uniform It is worthwhile to examine Maxwell s equations as they apply to a plane wave and see if they are satisfied, meaning that plane waves are consistent with the mathematical framework As I mentioned earlier, we will be discussing the propagation of EM waves in empty space (a vacuum). This means that Q enc = 0 in Gauss s law for electric fields and I enc = 0 in Ampere s law. To apply the first two of Maxwell s equations we choose as our gaussian surface a closed rectangular box shown in the figure below on the right. B da = 0 is always true for any magnetic field. The box does not enclose any charge, so E da = 0. E da everywhere on the gaussian surface except the top and bottom. E is uniform so the flux thru the top surface is +EA and thru the bottom surface is EA. The net flux over the entire gaussian surface is zero and this equation is satisfied. If we enclosed the wave front shown below on the left such that part of the box lies in the region where E = 0, then if the electric field has a component parallel to the x-axis (i.e., E x 0), there will be zero flux thru the right surface but not thru the left and therefore E da = 0 will not be satisfied. Same goes for the magnetic field. This means that the electric and magnetic fields must be
perpendicular to the direction of propagation (in this case, the x-axis) EM waves are transverse Next consider Faraday s law written out explicitly: E dl = dφ B d B da The closed path that is chosen for the line integral defines the boundary of the surface used to compute the magnetic flux (note that this flux is not computed over a closed surface). According to the RHR, curl the fingers of your right hand in the direction used to compute the closed path integral (either CW or CCW) and your thumb will point in the general direction of the da vectors on the surface. Consider the closed loop efgh shown in the figures below where at the instant shown, the wave front is located somewhere between the sides perpendicular to the x-axis. This rectangle has length Δx and wih a. E = 0 to the right of the wave front and so does not contribute to the line integral. To the left of the wave front, E dl = 0 everywhere where the path is parallel to the x-axis and only along gh is the line integral nonzero. E and dl are in opposite directions on this segment of length a, therefore the line integral along gh is Ea and so E dl = Ea. Now let s compute the right hand side of Faraday s law, the magnetic flux. We have assumed a magnetic field that points in the z-direction at this particular instant and by the RHR, da also points in the z-direction (out of =
the page). So, B is everywhere uniform and parallel to da on the flat surface defined by efgh. However, we don t need the flux but, rather, the change in flux. During a time-interval the wave front moves a distance c in the x- direction and the magnetic flux has increased by an amount dφ B = Ba(c ), so the rate of change of magnetic flux is dφ B = Bac. For a plane wave, Faraday s law becomes Ea = Bac. So, for a plane wave to satisfy Faraday s law, it must obey the relationship E = cb, where c is the speed of an electromagnetic wave. Finally, let s see if there are any further restrictions placed on plane EM waves by Ampere s law. There is no physical conduction current (I enc = 0) so dφ Ampere s law written out explicitly is: B dl = ε 0 μ E 0 = d ε 0 μ 0 E da. In order to calculate both integrals, we move our rectangular loop so it s in the xz-plane as shown in the figures below.
Using the same arguments as for the integrals in Faraday s law, with E and B reversed, we have B dl = Ba since this integral is zero everywhere except along gh, which has length a. The line integral is taken CCW around the loop so by the RHR, da points out of the page in (b) and is parallel to E. In a time-interval, the electric flux increases by dφ E = Ea(c ) and so dφ E = Eac For a plane wave, Ampere s law becomes Ba = ε 0 μ 0 Eac and so B = ε 0 μ 0 Ec is another condition that plane EM waves must satisfy in addition to E = cb required because of Faraday s law. Combining the two fixes the speed that plane EM waves must travel in a vacuum: c = 1 ε 0 μ 0. Numerically, c = 3.00 x 10 8 m/s. One other assumption that we made for this plane EM wave was that E B and it turns out that the direction of propagation is given by the vector E B
In the previous discussion, having the electric field point in the y-direction was completely arbitrary. The term polarization refers to the orientation of the EM fields as they oscillate. By convention, the electric field determines the polarization of an EM wave. If the oscillation of this field is along a particular direction, then the wave is said to be linearly polarized. If this direction rotates as the wave propagates thru space, then the wave is said to be circularly polarized or elliptically polarized. The Electromagnetic Wave Equation In CH 15, we derived a differential equation (the wave equation) that describes how the amplitude y of a mechanical wave varies in space and time: 2 y = 1 2 y x 2 v 2 t2, where v is the (constant) propagation speed of the wave It turns out that using Maxwell s equations for empty space, a wave equation can also be derived for the electric field as well as one for the magnetic field 2 E = ε x 0μ 2 E 2 0 and 2 B = ε t 2 x 0μ 2 B 2 0 t2. And this implies that all EM waves (not just plane waves) must travel at speed c = 1 in empty space ε 0 μ 0
Sinusoidal Electromagnetic Waves A sinusoidal EM wave mathematically behaves just like the sinusoidal transverse mechanical wave on a string that we studied in CH 15 E and B at a particular point in space are sinusoidal functions of time and, at a particular instant in time, the fields vary sinusoidally in space (the figure below shows a snapshot of a linearly polarized wave at some instant in time) Many sinusoidal EM waves are plane waves at any instant in time, the fields are uniform over any plane perpendicular to the direction of propagation but not uniform from one plane to the next For sinusoidally varying plane waves, the electric and magnetic fields oscillate in phase: E is zero where B is zero and E is maximum where B is maximum and, as before, the direction of propagation is given by the vector E B. If we take the direction of propagation to be in the +x-direction and let the y- axis correspond to the direction that the electric field is oscillating, then the magnetic field oscillates along the z-axis. Let E max and B max denote the amplitudes (maximum field strength) of the oscillations of the two fields. Then the wavefunctions for the fields comprising the EM wave are: E y (x, t) = E max cos(kx ωt) and B z (x, t) = B max cos(kx ωt). Now that the fields are no longer uniform in space, it is the amplitudes that must satisfy the restriction imposed by Faraday s law: E max = cb max
Written as vectors, the wavefunctions above would be written as E (x, t) = j E max cos(kx ωt) and B (x, t) = k B max cos(kx ωt). E B ~ (j k ) = i, which is the direction of propagation. For an EM wave traveling in the x-direction, the argument of the cosine function changes to kx + ωt, as we learned in CH 15 for sinusoidal transverse mechanical waves. For transverse EM waves, we also have to ensure that E B ~ i. By the RHR, j ( k ) = i and ( j ) k = i. So having E (x, t) ~ j and B (x, t) ~ k OR E (x, t) ~ j and B (x, t) ~ k will work. The theory behind the behavior of EM waves in the presence of matter (as opposed to empty space) can be considerably more complicated and I am not going to cover this **EXAMPLE 32.1 1061** Electromagnetic Energy Flow and the Poynting Vector Just like mechanical waves, EM waves transport energy. In Chapter 24, we derived the energy density in an electric field and in Chapter 30, we derived the energy density in a magnetic field. So in a region devoid of matter, but where E and B fields are present, the total electromagnetic energy density is u = u E + u B = 1 2 ε 0E 2 + 1 2μ 0 B 2 Since B = E c ε 0 E 2 = 1 μ 0 B 2 and c = 1 ε 0 μ 0, u = 1 2 ε 0E 2 + 1 2μ 0 c 2 E2 = 1 2 ε 0E 2 + 1 2 ε 0E 2 = This shows that in a vacuum, the energy transported by an EM wave is split equally between the energy stored in the electric field and the energy stored in the magnetic field. Also, the fields oscillate in both position and time, which means the energy density does too. In Chapter 15, we discussed both traveling waves, which transport energy from one region of space to another, as well as standing waves, which do not. For traveling waves, we quantified the flow of energy by defining the intensity of the wave. This is the amount of energy transferred thru a cross-
sectional area perpendicular to the direction of propagation per unit time. In other words, the wave s power per unit area. The variable S will be used to represent intensity and has units W/m 2. We can compute the intensity of a plane EM wave by considering a rectangular box of thickness dx and cross-sectional area A, where the rectangular face is perpendicular to the direction of propagation. If dx is sufficiently small, the fields are approximately uniform within the box and the total energy inside is the sum of the electric and magnetic energy densities multiplied by the volume of the box. So, du = (u E + u B )Adx = 1 (ε 2 0E 2 + B2 ) Adx μ 0 The energy is moving at the speed of light, so all of the energy contained in the box exits it in a time = dx/c. Therefore, the rate at which energy is transferred thru the cross-sectional area A is du = 1 (ε 2 0E 2 + B2 ) Adx = μ 0 dx/c c (ε 2 0E 2 + B2 ) A. The intensity (rate of energy flow per unit area, du/ ) is μ 0 A S = c 2 (ε 0E 2 + B2 μ 0 ) = cε 0 E 2. Since E = cb and c = 1, we can also express the intensity as S = ε 0 μ 0 cε 0 E 2 = ε 0 c 2 BE = EB. The direction of this energy flow is in the direction μ 0 of propagation (E B ) and the intensity can be defined as a vector quantity. This is called the Poynting vector: S = 1 μ 0 (E B ) The vector S Poynts (actually, Poynting is the name of the physicist who proposed this idea) in the direction of propagation of the EM wave and has magnitude equal to EB μ 0 since E B. The instantaneous intensity is an oscillating function of time but often we are not interested in this rapid oscillation but, instead, want the average intensity S av. Since the instantaneous intensity is a product of sinusoidally varying terms that are in phase, the average intensity is just half the value of the maximum (or peak) intensity.
= Emax 2 2μ 0 c Thus, S av = 1 (EB) μ av = E maxb max 0 2μ 0 E = cb was used to obtain the last two equalities = cb 2 max, where the relationship 2μ 0 **EXAMPLE 34-4 905** **EXAMPLE 34-5 906** Electromagnetic Momentum Flow and Radiation Pressure In addition to transporting energy, electromagnetic waves also carry momentum with them. It will take us a bit far afield to derive the following, so you will have to, for now, accept the fact that the energy and momentum in an EM wave are related as: U = pc and therefore, du dp = c. By definition, the rate of energy flow per unit area is S = 1 du so the rate of A momentum flow per unit area is 1 dp = S = EB. A c cμ 0 From Newton s 2 nd law, we know that dp = F and pressure is defined as a force per unit area. The average force per unit area for an EM wave is called radiation pressure. That is, p rad = 1 A (dp ) av If the EM wave is completely absorbed, then p rad = S av c However, if the wave is completely reflected, then the change in momentum is twice as great and p rad = 2 S av c Lasers exert enough electromagnetic pressure to levitate small objects. Radiation pressure has been suggested as a means of driving spacecraft by using solar sails. IKAROS, launched in 2010 by JAXA (Japan s space agency), was the first practical solar sail vehicle. As of 2015, it was still under acceleration, proving the practicality of a solar sail for long-duration missions. The idea that EM waves carry momentum played an important role in Einstein s development of his famous equation E = mc 2 **EXAMPLE 32.5 1067**