k;2
l p Ch. 6 Quantization of Electromagnetic Radiation 186
E-Field Vector
Wave Number
B-Field Vector
Angular Frequency
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(6.6)
There are a couple of points that are important to realize about Equations 6.2 and 6.4 in relationship to Figure 6.3. The first point is that the E and B field components of the electromagnetic wave are in phase with each other in space and time. This can be visualized by realizing that as time goes on the entire field structure of Figure 6.3 moves along as a unit at the speed c. If the wave moves past a point in space, however, the electric and magnetic fields at that point change in phase every instant, with both E and B attaining their maximum or minimum at the same point in space and at the same instant in time. The second point to be realized is that Figure 6.3 represents a plot of E versus the position coordinate x at a constant value of time, say t 5 t0; 0. Thus E(x, t0) has a sinusoidal de-pendence on x with a wavelength l 5 2p/k. Likewise, if x is held constant, say x 5 x0; 0, a plot of E(x0, t) versus t would look like Figure 6.3, with the X-axis being replaced by a t-axis. In this case, the period of oscillation (instead of wavelength) would be given by T 5 1/n 5 2p/v.
Before leaving this section, it should be emphasized that the qualita-tive discussion of the origin of electromagnetic waves has been concerned with only those waves produced by linear acceleration of an electric charge.
However, electromagnetic radiation occurs whenever an electric charge is accelerated, irrespectively of the manner in which it is accelerated. For ex-ample, a charge in uniform circular motion experiences centripetal accel-eration that produces a circularly polarized electromagnetic wave. Such waves are commonly produced by a synchrotron, which imparts very high
2 .
5 5 5
c ln 2 k
p
l pn v
6.1 Properties and Origin of Electromagnetic Waves 187
Y
Z
E
B
E
B
c X
2/C 4/C D
Em l
Bm B(x, to)
E(x, to)
Figure 6.3
An electromagnetic plane polarized mono-chromatic traveling wave, with the trans-verse E-field vectors in the XY plane.
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speeds to charged particles by a high-frequency electric field combined with a low-frequency magnetic field. As a last point of interest, we can infer from this section that the frequency of an electromagnetic wave pro-duced by an accelerating charge depends on the frequency of oscillation of that charge. Conversely, an electric charge, say the electrons in a receiv-ing antenna, will be accelerated by the forces they encounter from passreceiv-ing electromagnetic waves. The frequency of the resulting alternating current will then depend on the frequency of the incident electromagnetic waves.
6.2 Intensity, Pressure, and Power of Electromagnetic Waves
The laws of electricity, magnetism, optics, and the propagation of electro-magnetic waves were well understood by 1864 and completely contained in a set of four partial differential equations—known as Maxwell’s equa-tions. Although it would not serve our objectives to develop James Clark Maxwell’s electrodynamics, one of his equations in differential form, namely
(6.7)
will be briefly utilized. This equation is normally derived in general physics starting with Faraday’s law of induction and invoking the calculus of Stokes’ formula. The inverted delta symbol =, as given in Appendix A Sec-tion A.9, is called the del operator and defined by
(6.8)
The curly dees in the expression −/−x are, as you may well know, simply interpreted as the partial derivative with respect to x. We need only consider the operational nature of Equations 6.7 and 6.8 and their application to the plane-polarized electromagnetic wave of the previous section. That is, direct substitution of Equations 6.2 and 6.4 into Equation 6.7 gives
Emk cos (kx 2 vt)k 5 2Bm(2v) cos (kx 2 vt)k. (6.9) The left-hand side of this equation is directly obtained from the curl of E (i.e., = 3 E), when it is realized that for E 5 E(x, t) the partials with respect
3E5 B, 2t
= 22
1 1 .
; i j z k
x y
2 2
2 2
2
= 2
Ch. 6 Quantization of Electromagnetic Radiation 188
Maxwell’s Equation
Del Operator
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6.2 Intensity, Pressure, and Power of Electromagnetic Waves 189
to y and z become identically zero, and the remaining term, (−E/−x) (i 3 j), gives a vector pointing in the k-direction (i.e., i 3 j 5 k). Equation 6.9 im-mediately reduces to
Em5 cBm (6.10)
by substitution from Equation 6.6. Further, considering only the magni-tudes of E and B given in Equations 6.2 and 6.4, the ratio of these two equations gives
which combines with Equation 6.10 to give
E 5 cB (6.11)
for the instantaneous magnitudes of E and B. Thus taking into account the vector properties of E, B, and c we obtain
E 5 B 3 c. (6.12)
The results given by Equations 6.10 to 6.12 show the interdependence of the electric and magnetic field vectors, and they will prove most useful in developing equations for the energy transported by an electromagnetic wave. Usually, the energy transmitted in a radiation field by an electro-magnetic wave is specified in terms of the intensity, which can be simply thought of as energy per unit area per unit of time. More specifically, in-tensity is the energy per unit time transmitted across a unit area that is nor-mal to the direction of propagation of a wave. It can be calculated using the Poynting vector S, which is defined in general physics by the equation
(6.13)
The quantity m0in this equation is called the permeability constant, which has the defined value (exactly) of
(6.14) 5 ,
B E
B E
m m
; 3
S 1 E B m0
4 310 .
; A
7 N
2
0 2
m p
Poynting Vector Plane Waves
Permeability Constant
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The units of S are easily obtained by realizing that E has units of N/C 5
as expected for a quantity representing energy per unit time per unit area.
It should also be realized that the magnitude of S divided by the speed of light c would give the amount of radiant energy per unit volume of space, which is called the energy density. That is, using the symbol e to represent electromagnetic wave energy to distinguish it from particle energy, S could be thought of as
(6.16)
where A represents the unit surface area. Now, since the electromagnetic wave is propagating in the x-direction with a speed c 5 dx/dt, then
(6.17a)
Frequently, the ratio of S to c is called the radiation pressure, since 9, Ch. 6 Quantization of Electromagnetic Radiation 190
Magnetic Induction
Instantaneous Intensity
Energy Density
Radiation Pressure (6.17b)