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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 𝑐2 = 𝜀0 𝜇0 . 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 108 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 Qenc = 0 in Gauss’s law for electric fields and Ienc = 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. ⃗⃗ ⋅ 𝑑𝑨 ⃗⃗ = 0 is always true for any magnetic field. The box does not ∮𝑩 ⃗⃗ ⋅ 𝑑𝑨 ⃗⃗ = 0. 𝑬 ⃗⃗ ⊥ 𝑑𝑨 ⃗⃗ everywhere on the gaussian enclose any charge, so ∮ 𝑬 ⃗⃗ is uniform so the flux thru the top surface except the top and bottom. 𝑬 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., Ex ≠ 0), there will be zero flux thru the right surface but not thru ⃗⃗ ⋅ 𝑑𝑨 ⃗⃗ = 0 will not be satisfied. Same goes for the the left and therefore ∮ 𝑬 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 ⃗ ⋅ 𝑑𝒍 = − Next consider Faraday’s law written out explicitly: ∮ ⃗𝑬 − 𝑑 𝑑𝑡 𝑑Φ𝐵 𝑑𝑡 = ⃗⃗ ⋅ 𝑑𝑨 ⃗⃗ ∫𝑩 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 ⃗⃗ vectors on the surface. the 𝑑𝑨 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 width a. E = 0 to the right of the wave front and so does not contribute to the line integral. ⃗⃗ ⋅ 𝑑𝒍 = 0 everywhere where the path is To the left of the wave front, 𝑬 ⃗⃗ and 𝑑𝒍 parallel to the x-axis and only along gh is the line integral nonzero. 𝑬 are in opposite directions on this segment of length a, therefore the line ⃗ ⋅ 𝑑𝒍 = −𝐸𝑎. integral along gh is –Ea and so ∮ ⃗𝑬 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 ⃗⃗ also points in the z-direction (out of particular instant and by the RHR, 𝑑𝑨 ⃗⃗ is everywhere uniform and parallel to 𝑑𝑨 ⃗⃗ on the flat surface the page). So, 𝑩 defined by efgh. However, we don’t need the flux but, rather, the change in flux. During a time-interval dt the wave front moves a distance c∙dt in the xdirection and the magnetic flux has increased by an amount 𝑑Φ𝐵 = 𝐵𝑎(𝑐 ∙ 𝑑𝑡), so the rate of change of magnetic flux is 𝑑Φ𝐵 𝑑𝑡 = 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 (Ienc = 0) so ⃗ ⋅ 𝑑𝒍 = 𝜀0 𝜇0 𝑑Φ𝐸 = Ampere’s law written out explicitly is: ∮ ⃗𝑩 𝑑 𝑑𝑡 ⃗⃗ ⋅ 𝑑𝑨 ⃗⃗ . In order to calculate both integrals, we move our ∫𝑬 rectangular loop so it’s in the xz-plane as shown in the figures below. 𝜀0 𝜇0 𝑑𝑡 ⃗ and Using the same arguments as for the integrals in Faraday’s law, with ⃗𝑬 ⃗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 ⃗⃗ points out of the page in (b) and is parallel to 𝑬 ⃗⃗ . the loop so by the RHR, 𝑑𝑨 In a time-interval dt, the electric flux increases by 𝑑Φ𝐸 = 𝐸𝑎(𝑐 ∙ 𝑑𝑡) and so 𝑑Φ𝐸 𝑑𝑡 = 𝐸𝑎𝑐 For a plane wave, Ampere’s law becomes 𝐵𝑎 = 𝜀0 𝜇0 𝐸𝑎𝑐 and so 𝐵 = 𝜀0 𝜇0 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 √ 𝜀0 𝜇 0 . Numerically, c = 3.00 x 108 m/s. ⃗⃗ ⊥ 𝑩 ⃗⃗ 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 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 𝜕𝑥 2 = 1 𝜕2 𝑦 𝑣 2 𝜕𝑡 2 , 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 𝜕𝑥 2 = 𝜀0 𝜇0 𝜕2 ⃗𝑬 𝜕𝑡 2 and ⃗ 𝜕2 ⃗𝑩 𝜕𝑥 2 = 𝜀0 𝜇0 ⃗ 𝜕2 ⃗𝑩 𝜕𝑡 2 . And this implies that all EM waves (not just plane waves) must travel at speed 𝑐 = 1 √ 𝜀0 𝜇 0 in empty space 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 ⃗ and ⃗𝑩 ⃗ at a particular point in space are sinusoidal functions of time and, at ⃗𝑬 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 ⃗ is zero where ⃗𝑩 ⃗ is zero and ⃗𝑬 ⃗ is maximum where ⃗𝑩 ⃗ is oscillate in phase: ⃗𝑬 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 yaxis correspond to the direction that the electric field is oscillating, then the magnetic field oscillates along the z-axis. Let 𝐸𝑚𝑎𝑥 and 𝐵𝑚𝑎𝑥 denote the amplitudes (maximum field strength) of the oscillations of the two fields. Then the wavefunctions for the fields comprising the EM wave are: 𝐸𝑦 (𝑥, 𝑡) = 𝐸𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) and 𝐵𝑧 (𝑥, 𝑡) = 𝐵𝑚𝑎𝑥 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: 𝐸𝑚𝑎𝑥 = 𝑐𝐵𝑚𝑎𝑥 ⃗⃗ (𝑥, 𝑡) = Written as vectors, the wavefunctions above would be written as 𝑬 ̂𝐵𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡). 𝑬 ⃗⃗ (𝑥, 𝑡) = 𝒌 ⃗⃗ × 𝒋̂𝐸𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) and 𝑩 ̂) = 𝒊̂, which is the direction of propagation. ⃗⃗ ~ (𝒋̂ × 𝒌 𝑩 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 ̂) = −𝒊̂ and (−𝒋̂) × 𝒌 ̂ = −𝒊̂. ⃗⃗ × 𝑩 ⃗⃗ ~ − 𝒊̂. By the RHR, 𝒋̂ × (−𝒌 ensure that 𝑬 ̂ OR ⃗𝑬 ̂ ⃗ (𝑥, 𝑡) ~ 𝒋̂ and ⃗𝑩 ⃗ (𝑥, 𝑡) ~ − 𝒌 ⃗ (𝑥, 𝑡) ~ − 𝒋̂ and ⃗𝑩 ⃗ (𝑥, 𝑡) ~ 𝒌 So having ⃗𝑬 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 ⃗ and ⃗𝑩 ⃗ fields are present, the total electromagnetic energy density where ⃗𝑬 1 1 2 2𝜇0 is 𝑢 = 𝑢𝐸 + 𝑢𝐵 = 𝜀0 𝐸 2 + Since 𝐵 = 𝜀0 𝐸 2 = 1 𝜇0 𝐸 𝑐 and 𝑐 = 1 √ 𝜀0 𝜇 0 𝐵2 1 , 𝑢 = 𝜀0 𝐸 2 + 2 1 2𝜇0 𝑐2 1 1 2 2 𝐸 2 = 𝜀0 𝐸 2 + 𝜀0 𝐸 2 = 𝐵2 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/m2. 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. 1 𝐵2 So, 𝑑𝑈 = (𝑢𝐸 + 𝑢𝐵 )𝐴𝑑𝑥 = (𝜀0 𝐸 2 + ) 𝐴𝑑𝑥 2 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 dt = dx/c. Therefore, the rate at which energy is transferred thru the cross-sectional area A is 𝑑𝑈 𝑑𝑡 𝐵2 𝑐 1 𝐵2 0 (𝜀0 𝐸 2 + 𝜇 ) 𝐴. The intensity (rate of energy flow per unit area, 2 0 𝑐 𝐴𝑑𝑥 = (𝜀0 𝐸 2 + ) = 2 𝜇 𝑑𝑥/𝑐 𝑑𝑈/𝑑𝑡 𝐴 ) is 𝐵2 𝑆 = (𝜀0 𝐸 2 + ) = 𝑐𝜀0 𝐸 2 . 2 𝜇 0 Since 𝐸 = 𝑐𝐵 and 𝑐 = 𝑐𝜀0 𝐸 2 = 𝜀0 𝑐 2 𝐵𝐸 = 𝐸𝐵 𝜇0 1 √ 𝜀0 𝜇 0 , we can also express the intensity as 𝑆 = . The direction of this energy flow is in the direction ⃗⃗ × ⃗𝑩 ⃗ ) and the intensity can be defined as a vector quantity. of propagation (𝑬 This is called the Poynting vector: ⃗𝑺 = 1 ⃗⃗ × ⃗𝑩 ⃗) (𝑬 𝜇0 ⃗ “Poynts” (actually, Poynting is the name of the physicist who The vector 𝑺 proposed this idea) in the direction of propagation of the EM wave and has 𝐸𝐵 ⃗ ⊥ ⃗𝑩 ⃗. magnitude equal to since ⃗𝑬 𝜇0 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 𝑆𝑎𝑣 . 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. Thus, 𝑆𝑎𝑣 = 1 𝜇0 (𝐸𝐵)𝑎𝑣 = 𝐸𝑚𝑎𝑥 𝐵𝑚𝑎𝑥 2𝜇0 = 𝐸𝑚𝑎𝑥 2 2𝜇0 𝑐 = 𝑐𝐵𝑚𝑎𝑥 2 2𝜇0 , where the relationship 𝐸 = 𝑐𝐵 was used to obtain the last two equalities **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, definition, the rate of energy flow per unit area is 𝑆 = momentum flow per unit area is 1 𝑑𝑝 𝐴 𝑑𝑡 𝑆 𝐸𝐵 𝑐 𝑑𝑝 𝑐𝜇0 = = From Newton’s 2nd law, we know that 𝑑𝑡 𝑑𝑈 =𝑐 𝑑𝑡 1 𝑑𝑈 𝐴 𝑑𝑡 𝑑𝑝 𝑑𝑡 . By so the rate of . = 𝐹 and pressure is defined as a force per unit area. The average force per unit area for an EM wave is called 1 𝑑𝑝 radiation pressure. That is, 𝑝𝑟𝑎𝑑 = ( ) 𝐴 𝑑𝑡 𝑎𝑣 If the EM wave is completely absorbed, then 𝑝𝑟𝑎𝑑 = 𝑆𝑎𝑣 𝑐 However, if the wave is completely reflected, then the change in momentum is twice as great and 𝑝𝑟𝑎𝑑 = 2 𝑆𝑎𝑣 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**