Electromagnetic Waves

Consider a plane electromagnetic wave propagating through a vacuum in the $z$-direction. Incidentally, electromagnetic waves are the only commonly-occurring waves that do not require a medium through which to propagate. Suppose that the wave is linearly polarized in the $x$-direction; that is, its electric component oscillates in the $x$-direction. It follows that the magnetic component of the wave oscillates in the $y$-direction (Fitzpatrick 2008). According to standard electromagnetic theory (see Appendix C), the wave is described by the following pair of coupled partial differential equations:

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon_0}\,\frac{\partial H_y}{\partial z},$$ (6.116)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z},$$ (6.117)

where $E_x(z,t)$ is the electric field-strength, and $H_y(z,t)$ is the magnetic intensity (i.e., the magnetic field-strength divided by $\mu_0$). Observe that Equations (6.116) and (6.117), which govern the propagation of electromagnetic waves through a vacuum, are analogous to Equations (6.53) and (6.54), which govern the propagation of electromagnetic signals down a transmission line. In particular, $E_x$ has units of voltage over length, $H_y$ has units of current over length, $\epsilon_0$ has units of capacitance per unit length, and $\mu_0$ has units of inductance per unit length.

Equations (6.116) and (6.117) can be combined to give

$$\frac{\partial^{\,2} E_x}{\partial t^{\,2}} =\frac{1}{\epsilon_0\,\mu_0}\,\frac{\partial^{\,2} E_x}{\partial z^{\,2}},$$ (6.118)

$$\frac{\partial^{\,2} H_y}{\partial t^{\,2}} =\frac{1}{\epsilon_0\,\mu_0}\,\frac{\partial^{\,2} H_y}{\partial z^{\,2}}.$$ (6.119)

It follows that the electric and the magnetic components of an electromagnetic wave propagating through a vacuum both separately satisfy a wave equation of the form (6.1). Furthermore, the phase velocity of the wave is the velocity of light in vacuum,

$$c = \frac{1}{\sqrt{\epsilon_0\,\mu_0}}=2.998\times 10^8\,{\rm m\,s}^{-1}.$$ (6.120)

Let us search for a traveling wave solution of Equations (6.116) and (6.117), propagating in the positive $z$-direction, whose electric component has the form

$$E_x(z,t)=E_0\,\cos(\omega\,t-k\,z-\phi).$$ (6.121)

This is a valid solution provided that

$$\omega=k\,c.$$ (6.122)

According to Equation (6.117), the magnetic component of the wave is written

$$H_y(z,t)=Z^{\,-1}\,E_0\,\cos(\omega\,t-k\,z-\phi),$$ (6.123)

where

$$Z= Z_0 \equiv \sqrt{\frac{\mu_0}{\epsilon_0}},$$ (6.124)

and $Z_0$ is the impedance of free space. [See Equation (6.74).] Thus, the electric and magnetic components of an electromagnetic wave propagating through a vacuum are mutually perpendicular, and also perpendicular to the direction of propagation. Moreover, the two components oscillate in phase (i.e., they have simultaneous maxima and zeros), and the amplitude of the magnetic component is that of the electric component divided by the impedance of free space.

Multiplying Equation (6.116) by $\epsilon_0\,E_x$, Equation (6.117) by $\mu_0\,H_y$, and adding the two resulting expressions, we obtain the energy conservation equation

$$\frac{\partial{\cal E}}{\partial t} + \frac{\partial{\cal I}_z}{\partial z} =0,$$ (6.125)

where

$${\cal E} = \frac{1}{2}\left(\epsilon_0\,E_x^{\,2} + \mu_0\,H_y^{\,2}\right)$$ (6.126)

is the energy density (i.e., energy per unit volume) of the wave (Fitzpatrick 2008), whereas

$${\cal I}_z = E_x\,H_y$$ (6.127)

is the energy flux (i.e., power per unit area) in the positive $z$-direction. (See Appendix C.) The mean energy flux associated with the $z$-directed electromagnetic wave specified in Equations (6.121) and (6.123) is thus

$$\langle {\cal I}_z \rangle = \frac{1}{2}\,\frac{E_0^{\,2}}{Z}.$$ (6.128)

For a similar wave propagating in the negative $z$-direction, it can be demonstrated that

$$E_x(z,t) =E_0\,\cos(\omega\,t+k\,z-\phi),$$ (6.129)

$$H_y(z,t) =-Z^{\,-1}\,E_0\,\cos(\omega\,t+k\,z-\phi),$$ (6.130)

and

$$\langle {\cal I}_z \rangle =- \frac{1}{2}\,\frac{E_0^{\,2}}{Z}.$$ (6.131)

Consider a plane electromagnetic wave, linearly polarized in the $x$-direction, that propagates in the $z$-direction through a transparent dielectric medium, such as glass or water. As is well-known (Fitzpatrick 2008), the electric component of the wave causes the neutral molecules making up the medium to polarize; that is, it causes a small separation to develop between the mean positions of the positively and negatively charged constituents of the molecules (i.e., the atomic nuclei and the orbiting electrons). [Incidentally, it can be shown that the magnetic component of the wave has a negligible influence on the molecules, provided the wave amplitude is sufficiently small that the wave electric field does not cause the electrons and nuclei to move with relativistic velocities (ibid.).] If the mean position of the positively charged constituents of a given molecule, of net charge $+q$, develops a vector displacement ${\bf d}$ with respect to the mean position of the negatively charged constituents, of net charge $-q$, in response to a wave electric field ${\bf E}$, then the associated electric dipole moment is ${\bf p} = q\,{\bf d}$, where ${\bf d}$ is generally parallel to ${\bf E}$ (ibid.). Furthermore, if there are $N$ such molecules per unit volume then the electric dipole moment per unit volume is written ${\bf P} = N\,q\,{\bf d}$. In a linear, isotropic, dielectric medium (ibid.),

$${\bf P} = \epsilon_0\,(\epsilon-1)\,{\bf E},$$ (6.132)

where $\epsilon>1$ is a dimensionless quantity, known as the relative dielectric constant, that is a property of the medium in question. In the presence of a dielectric medium, Equations (6.116) and (6.117) generalize to give

$$\frac{\partial E_x}{\partial t} = - \frac{1}{\epsilon_0}\left(\frac{\partial P_x}{\partial t}+\frac{\partial H_y}{\partial z}\right),$$ (6.133)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (6.134)

(See Appendix C.) When combined with Equation (6.132), these expressions yield

$$\frac{\partial E_x}{\partial t} =- \frac{1}{\epsilon\,\epsilon_0}\,\frac{\partial H_y}{\partial z},$$ (6.135)

$$\frac{\partial H_y}{\partial t} = -\frac{1}{\mu_0}\,\frac{\partial E_x}{\partial z}.$$ (6.136)

It can be seen that the previous equations are just like the corresponding vacuum equations, (6.116) and (6.117), except that $\epsilon_0$ has been replaced by $\epsilon\,\epsilon_0$. It immediately follows that the phase velocity of an electromagnetic wave propagating through a dielectric medium is

$$v = \frac{1}{\sqrt{\epsilon\,\epsilon_0\,\mu_0}} = \frac{c}{n},$$ (6.137)

where $c=1/(\epsilon_0\,\mu_0)^{1/2}$ is the velocity of light in vacuum, and the dimensionless quantity

$$n = \sqrt{\epsilon}$$ (6.138)

is known as the refractive index of the medium. Thus, an electromagnetic wave propagating through a transparent dielectric medium does so at a phase velocity that is less than the velocity of light in vacuum by a factor $n$ (where $n>1$). The dispersion relation of the wave is thus

$$\index{dispersion relation!of electromagnetic wave in dielectric medium} \omega = k\,v = \frac{k\,c}{n}.$$ (6.139)

Furthermore, the impedance of a transparent dielectric medium becomes

$$Z = \sqrt{\frac{\mu_0}{\epsilon\,\epsilon_0}} = \frac{Z_0}{n},$$ (6.140)

where $Z_0$ is the impedance of free space.

Incidentally, the signal that travels down a transmission line is a form of guided electromagnetic wave. It follows that if the space between the two conductors that constitute the line is filled with dielectric material of relative dielectric constant $\epsilon$ then the signal propagates down the line at the reduced phase velocity

$$v = \frac{c}{\sqrt{\epsilon}}.$$ (6.141)

This occurs because the dielectric material increases the capacitance per unit length of the line by a factor $\epsilon$, but leaves the inductance per unit length unchanged. (See Section 6.6.) For the same reason, the presence of the dielectric material decreases the impedance of the line by a factor $\sqrt{\epsilon}$. Hence, the impedance of a dielectric filled co-axial cable is [cf., Equation (6.78)]

$$Z = \frac{1}{2\pi\sqrt{\epsilon}}\,\ln\left(\frac{b}{a}\right)\,Z_0.$$ (6.142)

Here, $a$ and $b$ are the radii of the inner and outer conductors, respectively.

Suppose that the plane $z=0$ forms the interface between two transparent dielectric media of refractive indices $n_1$ and $n_2$. Let the first medium occupy the region $z<0$, and the second the region $z>0$. Suppose that a plane electromagnetic wave, linearly polarized in the $x$-direction, and propagating in the positive $z$-direction, is launched toward the interface from a wave source of angular frequency $\omega$ situated at $z=-\infty$. We expect the wave incident on the interface to be partly reflected, and partly transmitted. The wave electric and magnetic fields in the region $z<0$ are written

$$E_x(z,t) = E_i\,\cos(\omega\,t-k_1\,z) + E_r\,\cos(\omega\,t+k_1\,z),$$ (6.143)

$$H_y(z,t) =Z_1^{\,-1}\,E_i\,\cos(\omega\,t-k_1\,z)- Z_1^{\,-1}\,E_r\,\cos(\omega\,t+k_1\,z),$$ (6.144)

where $E_i$ is the amplitude of (the electric component of) the incident wave, $E_r$ the amplitude of the reflected wave, $k_1=n_1\,\omega/c$, and $Z_1=Z_0/n_1$. The wave electric and magnetic fields in the region $z>0$ take the form

$$E_x(z,t) = E_t\,\cos(\omega\,t-k_2\,z),$$ (6.145)

$$H_y(z,t) =Z_2^{\,-1}\,E_t\,\cos(\omega\,t-k_2\,z),$$ (6.146)

where $E_t$ is the amplitude of the transmitted wave, $k_2=n_2\,\omega/c$, and $Z_2=Z_0/n_2$. According to standard electromagnetic theory (see Appendix C), the appropriate matching conditions at the interface ($z=0$) are that $E_x$ and $H_y$ are both continuous. Thus, continuity of $E_x$ yields

$$E_i + E_r = E_t,$$ (6.147)

whereas continuity of $H_y$ gives

$$n_1\,(E_i-E_r) = n_2\,E_t,$$ (6.148)

because $Z^{\,-1}\propto n$. It follows that

$$E_r =\left(\frac{n_1-n_2}{n_1+n_2}\right)E_i,$$ (6.149)

$$E_t =\left(\frac{2\,n_1}{n_1+n_2}\right)E_i.$$ (6.150)

The coefficient of reflection, $R$, is defined as the ratio of the reflected to the incident energy flux, so that

$$R = \left(\frac{E_r}{E_i}\right)^2 = \left(\frac{n_1-n_2}{n_1+n_2}\right)^2.$$ (6.151)

Likewise, the coefficient of transmission, $T$, is the ratio of the transmitted to the incident energy flux, so that

$$T = \frac{Z_2^{\,-1}}{Z_1^{\,-1}}\left(\frac{E_t}{E_i}\right)^2= ... ..._2}{n_1}\left(\frac{E_t}{E_i}\right)^2 = \frac{4\,n_1\,n_2}{(n_1+n_2)^2} = 1-R.$$ (6.152)

It can be seen, first of all, that if $n_1=n_2$ then $E_r=0$ and $E_t=E_i$. In other words, if the two media have the same indices of refraction then there is no reflection at the interface between them, and the transmitted wave is consequently equal in amplitude to the incident wave. On the other hand, if $n_1\neq n_2$ then there is always some reflection at the interface. Indeed, the amplitude of the reflected wave is roughly proportional to the difference between $n_1$ and $n_2$. This has important practical consequences. We can only see a clean pane of glass in a window because some of the light incident on an air/glass interface is reflected, as a consequence of the different refractive indices of air and glass. As is well-known, it is a lot more difficult to see glass when it is submerged in water. This is because the refractive indices of glass and water are quite similar, and so there is very little reflection of light incident on a water/glass interface.

According to Equation (6.149), $E_r/E_i<0$ when $n_2> n_1$. The negative sign indicates a $\pi$ radian phase shift of the (electric component of the) reflected wave, with respect to the incident wave. We conclude that there is a $\pi$ radian phase shift of the reflected wave, relative to the incident wave, on reflection from an interface with a medium of greater refractive index. Conversely, there is zero phase shift on reflection from an interface with a medium of lesser refractive index. (This effect is important in thin-film interference. See Section 10.5.)

Equations (6.149)–(6.152) are analogous to Equations (6.87)–(6.90), with the inverse of the refractive index playing the role of impedance. This suggests, by analogy with earlier analysis, that we can prevent reflection of an electromagnetic wave normally incident at an interface between two transparent dielectric media of different refractive indices by separating the media in question by a thin transparent layer whose thickness is one quarter of a wavelength, and whose refractive index is the geometric mean of the refractive indices of the two media. This is the physical principle behind the non-reflective lens coatings used in high-quality optical instruments. (See Exercise 15.)