14.1Introduction

The key work in the study of electromagnetics of moving media is the Lorentz transformation (LT) law. Under these transformation laws, Maxwell’s equation has the same form in two reference frames moving with respect to each other with a uniform velocity v₀. The velocity of light in free space remains the same value c = 3 × 10⁸ m/s irrespective of the velocity of the observer measuring the velocity of light. The postulate of special velocity that the physical laws are form invariant among uniformly moving observers is consistent with LTs. A vast amount of literature is available, out of which the author has selected [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] as a sample. In this chapter, we study the electromagnetic wave interactions with a moving media. The moving frame of reference in which the medium is at rest is assumed to be inertial [1]; it does not accelerate and it does not rotate.

14.2Snell’s Law

Let us consider the oblique incidence of a plane s-wave discussed in Section 2.13 for the case of a stationary medium. Figure 14.1 shows the same geometry, except that the medium is now moving with a velocity v₀ along the positive z-axis as shown in Figure 14.1. Before we discuss Snell’s law for this case, let us revisit the question of the stationary medium and show that the frequency of the reflected wave will be the same as the frequency of the incident wave. The above-mentioned property as well as Snell’s law θ_i = θ_r follows from the satisfaction of the boundary condition at z = 0, for all x and t.

Let

where

c is the velocity of light in free space

ω_i is the frequency of the incident wave

ω_r is the frequency of the reflected wave

The boundary condition at PEC wall z = 0 is that E_tan = 0 for all x and t:

Since the y component is the tangential component, the left-hand side (LHS) of (14.3) is obtained by adding (14.1) and (14.2) and substituting z = 0:

Since (14.4) is not an equation for x and t but an identity in these two variables

Let us now study the case where v₀ ≠ 0 but v₀ ≪ c. We are considering “nonrelativistic” velocities. Thus, we consider the so-called Galilean transformations (GTs). Assuming that the boundary is z = 0, at t = 0, at any other time

In this case, the boundary condition E_tan = 0 is implemented at z = v₀t. Substituting this value in (14.1) and (14.2),

Since (14.7) is an identity in x as well as t, we get

Denoting

Equation 14.8 can be written as

Of course, β = 0 gives the result of (14.5a) and (14.5c) for the stationary boundary. When β ≠ 0, the frequency of the reflected wave is different from the frequency of the incident wave. This frequency difference is the basis of “Doppler radar,” which is used to measure the speed of the moving conducting object, for example, the speed at which you are driving the car. When the coefficient of t and x are the same in the two terms on the LHS of (14.7), the cosine term can be cancelled and we get

We still have total reflection from the conductor (PEC) even if it is moving. In this section, we showed that Snell’s law is obtained by implementing the boundary conditions at the interface for all points on the boundary (any x in the example) and for all times (any t in the example).

14.3Galilean Transformation

Let ∑ be the laboratory frame of reference and ∑′ be the frame of reference in which the moving medium is stationary as shown in Figure 14.2.

In the usual formulation, where t is “absolute,”

Let us now consider Faraday’s law in differential form for a moving medium.

Faraday’s law in integral form is usually stated in the form

or

where C is a closed curve bounding the open surface S. Figure 14.3 describes the geometry to help identify the various terms in (14.17).

The last term on the right-hand side (RHS) of (14.16) gives the total electromotive force (emf) induced in the closed circuit C, which could be changing with time, also the point function B could as well be changing with time. Equation 14.17 separates the contributions due to time-changing magnetic flux density field [called transformer emf, the first term on the RHS of (14.17)] and due to time-varying circuit C [motional emf, the second term on the RHS of (14.17)]. The last term on the RHS of (14.17) can be changed to $-{\displaystyle {\oint }_{c}B}\cdot \left(v\times \text{d}l\right)$. The term v × dl is the area swept by the element in 1 s. The last term is thus equal to −dΦ_m/dt due to the motion.

Applying Stoke’s theorem, the closed line integrals in (14.17) can be converted to surface integrals:

Since (14.18) is true for any S, provided it is the same S on all the integrals in (14.18), the integrands must be the same on both sides of (14.18).

Equation 14.18 is the same as (1.61) except that we used in (14.18) the symbol E′ instead of E to emphasize that the electric field in the definition of the emf for a moving circuit has to be the effective electric field. The expression for E′ can be obtained from (1.6) from its definition as the force for unit charge and is given by

Substitution of (14.20) in (14.19) results in (1.1), the differential form of Faraday’s law in the laboratory frame.

We can also show [10] that (14.20) and (14.22) are the consequence of imposing Galilean invariance of the form of Faraday’s law in differential form, that is,

Similarly, we can show [10] that the form invariance of Ampere’s law (1.2) under GT gives us the following additional relationships among the fields in inertial and laboratory frames:

There is some inconsistency among Equations 14.20 and 14.22, 14.23 and 14.24. For example, let us apply these equations to free space and show the inconsistency. From (14.20),

Equation 14.25 is in variance with (14.23).

Similarly, from (14.23),

Equation 14.26 is in variance with (14.22). Note that they become the same if c is taken as infinity. We have to reject the GT of Maxwell’s equations for moving media even for the nonrelativistic velocities of the medium. We show next that LTs given in the next section overcome these difficulties. For nonrelativistic velocities we can then formulate first-order Lorentz transformations (FOLT) as consistent approximations [4] discussed in Section 14.18.

GT does not alter the Newton’s force equation since

since

However, under LTs, to be discussed in the next section, we show that t ≠ t′, v ≠ v′ + v₀, m ≠ m′, F ≠ F′, and that GT is valid only for nonrelativistic velocities. For arbitrary value of the parameter β = v₀/c, one needs to consider relativistic mechanics based on LTs. LTs are connected with the theory of special relativity and a four-dimensional space.

14.4Lorentz Transformation

Since t = t′, in the GT, this transformation leaves the distance between two physical points unchanged in all coordinate systems.

The velocity of light as measured by a moving observer on z axis (at rest in ∑′) will be (c − v₀) for the geometry shown in Figure 14.2. However, experiments show that the velocity of light as measured by a stationary observer or a moving observer will remain the same. This first postulate of the theory of special relativity is well confirmed. LTs in four-dimensional space x, y, z, ict will preserve the first postulate.

Let us, for convenience, use the nomenclature x_μ, μ = 1, 2, 3, 4 to designate the coordinates in the four-dimensional space:

Note that we write the time component with the subscript 4 and designate it as imaginary. This should be carefully distinguished from [4] the imaginary notation in “quantum theory and wave theory” in some books. An alternative definition of x₄ without i in its definition is discussed in Section 14.19.

The distance between two points in the four-dimensional space will remain the same provided

For the geometry shown in Figure 14.2,

Equation 14.36 will be satisfied if

Moreover, in the limit v₀ → 0

The origin of ∑′ moves with a velocity v₀ with respect to the origin ∑. Thus, we have the position of the origin of ∑′ as z = vt. Substituting z′ = 0 in (14.39),

Solving the relevant equations with signs chosen to agree with (14.42), we obtain

Thus, we have

where

Generalization of the above, where the relative velocity between ∑ and ∑′ is v₀ instead of $\widehat{z}{v}_0$, is straightforward and is given by

Equation 14.50 can be written as

Equations 14.46 and 14.47 give rise to the length contraction and time dilation

In (14.52) [3] L = z₂ – z₁, where z₂ and z₁ are the instantaneous coordinates of the end points of the rod, observed at the same time t; $L_0=z_2^{\prime }-z_1^{\prime }$, where $z_2^{\prime }$ and $z_1^{\prime }$ are the coordinates at the end points in ∑′.

Equation 14.53b [3] is interpreted as time dilation. A clock moving relative to an observer is found to run more slowly than one at rest relative to the observer. Considering unstable elementary particles as a clock, one can explain the time dilation concept by examining the lifetime of the particles. In (14.53), τ₀ is the lifetime of the particle at rest in the system ∑′. The particle is moving with a uniform velocity v₀ relative to the system ∑. When viewed from ∑, the moving particle lives longer than a particle at rest in ∑. The “clock” in motion is observed to run more slowly than an identical one at rest. Reference [2] has a number of graphs and examples to illustrate length contraction and time dilation.

14.5Lorentz Scalars, Vectors, and Tensors

Equation 14.36 is a constraint involved in the rotation of coordinates in the four-dimensional space. R is an invariant under LTs. LTs are rotations in a four-dimensional Euclidean space. This orthogonal transformation can be written as

For the geometry of Figure 14.2, the transformation matrix is given by

Figure 14.4a shows the geometry of rotation of the coordinates in the x₃ – x₄ plane by an angle ψ that achieves the LT.

An inverse transformation is obtained by reversing the sign of β, which gives a_νμ. Thus we have

We also have

Equations 14.54a and 14.54b show that LT is an orthogonal transformation.

From the geometry (Figure 14.4), OC = OA + AC, giving

Similarly,

Comparing (14.56) with (14.54c), we get

Note that the angle ψ is complex and its cosine, cos ψ = γ ≥ 1. Thus, LT as a rotation, by an angle ψ, is a concept rather than a real angular rotation. Figure 14.4b shows L, L₀, T, T₀. The distance L₀ in the frame ∑′ is observed as L in the frame ∑ at constant time in ∑. Though L appears larger than L₀ in the figure, because of the complex nature of ψ,

The time interval T₀ in the frame ∑′ is seen as T in the frame ∑, where

A quantity that does not change under LT is called a Lorentz scalar, say Φ. Its derivative (can be called four-gradient) with respect to $x_{\mathrm{\mu }}^{\prime }$ is a four-vector.

The derivative transforms like a four-vector.

The four-divergence of a four-vector is Lorentz invariant:

If A_μ = ∂Φ/∂x_μ, (14.61) shows that the four-dimensional Laplacian operator is a Lorentz invariant operator.

where ▫′ is the vector operator in four dimensions in (14.61) and ▫² =  ▫ ⋅ ▫ in (14.62).

The scalar product of two four-vectors is a scalar invariant

Lorentz scalar can be considered as a tensor of zero rank and four-vector as a tensor of rank one.

A second-rank tensor is a set of 16 quantities that transforms as

An example of such a tensor is the field strength four-tensor F_μν, which can be used to combine two of the Maxwell equations, as shown in the next section.

14.6Electromagnetic Equations in Four-Dimensional Space

In the four-dimensional space,

One can write the electromagnetic equations in terms of suitably chosen scalars, four-vectors, and four-tensors of second rank. The quantity in four-dimensional space will be denoted by an italic symbol with a subscript μ. Also a shorthand notation, that a repeated Greek letter subscript implies summation over that subscript as it varies from 1 to 4, will be used.

Let us start with the continuity equation (1.5)

Let

Equation 1.5 is

Since μ is a Greek symbol that is repeated, we can use the shorthand notation given as the last term in (14.67).

Moreover, if we definite the four-vector differential operator ▫

the continuity equation can be written, in terms of four divergence, as

where the bold italic J_μ is a four-vector current density.

Let us next consider the Lorentz condition, when the medium is free space, from (1.21):

Let us consider the four-vector potential A_μ:

Equation 14.71 can be written as

The wave equation in free space

can be written as

Note that (14.73b) is obtained from

If the source is a point charge Q moving with a relativistic velocity v along a specified trajectory, the scalar and vector potentials, called Lienard–Wiechert potentials, can be calculated and discussed in Appendix 14G. We touched on this topic in Chapter 13 also.

The field strength tensor F_μν can be constructed so that the equations

are satisfied. For example, from (14.75) and (14.76) we can write

If (14.75) and (14.76) can be written (in the form of a four-dimensional curl operation) as

then (14.77) is (14.78) with μ = 3, ν = 2, and F_μν = F₃₂ =  − B₁ =  − B_x.

One can verify that (14.75) and (14.76) become (14.78) if F_μν, a 4×4 matrix, has the following elements expressed in terms of the components of E and B.

Though we only verified the element F₃₂ =  − B₁, one can check the correctness of the other elements.

With this field strength tensor, we can show that the two Maxwell equations in free space

take the form, in Lorentz space,

The two Maxwell equations, valid in any medium,

can be expressed in Lorentz space

For this reason F_μν is called field strength tensor. From (14.79), we note it is an antisymmetric tensor. The other two Maxwell equations, valid in any medium

can be combined as [10]

where

Since G_μν is a second-rank tensor, it transforms as

It is an antisymmetric tensor and can be called as excitation tensor.

14.7Lorentz Transformation of the Electromagnetic Fields

The field strength F_μν is a tensor; its transformation is given by (14.62).

The inverse transformation is obtained by interchanging primed and unprimed quantities and replacing β by −(β). For a general velocity υ_o, one can generalize (14.90)–(14.91) by

14.8Frequency Transformation and Phase Invariance

A four-vector wave number k_μ can be constructed

so that

Since (14.93b) is a Lorenz scalar, it is Lorentz invariant giving the concept of phase invariance. Thus, we obtain (14.94)

Equation 14.94 leads to the results given by (14.95):

where θ and θ′ are the angles of k and k′ relative to the direction of υ_o. Equation 14.95a is the relativistic formula for the frequency shift, called relativistic Doppler effect. Equation 14.95b is the relativistic formula for aberration [4]. Equations 14.95 can be more easily seen by considering the motion of the frame to be on the z-direction, and the wave is propagating in the x–z plane.

Dividing (14.96b) by (14.96d), we obtain (14.95b).

From (14.94), we obtain

which is the same as (14.95a).

14.9Reflection from a Moving Medium

Let us solve the problem of determining the power reflection coefficient α of a moving medium. We will use the LT technique to solve the problem. The incident wave will be transformed to the primed frame in which the medium 2 is at rest, and this reflection problem in the rest frame with a stationary boundary can be solved using the techniques developed in Chapter 2. After obtaining the primed reflection coefficient R′, we will relate it to the unprimed R.

14.9.3Power Reflection Coefficient of a Moving Mirror for s-Wave Incidence

If the medium 2 is a PEC, $R_s^{\prime }=-1$, and the power reflection coefficient α is given by

(14.109a)$$\mathrm{\alpha }={\left|R_s\right|}^2\frac{cos{\mathrm{\theta }}_r}{cos{\mathrm{\theta }}_i}=\frac{1}{{\mathrm{\beta }}^{\prime }}{\left|R_s\right|}^2$$

(14.109b)$$\mathrm{\alpha }=\frac{1}{{\mathrm{\beta }}^{\prime }}{\left[\frac{1-\mathrm{\beta }C}{1+\frac{\mathrm{\beta }C}{{\mathrm{\beta }}^{\prime }}}\right]}^2.$$

Simplifying (14.109b), the power reflection coefficient for the case of s-wave incidence on a PEC moving perpendicular to the interface is given by

(14.110)$$\mathrm{\alpha }=\frac{\left(1+{\mathrm{\beta }}^2-2\mathrm{\beta }C\right)\left(1+{\mathrm{\beta }}^2-\frac{2\mathrm{\beta }}{C}\right)}{{\left(1-{\mathrm{\beta }}^2\right)}^2}.$$

It is worth remarking that the PEC medium is a perfect reflector and α is the highest power reflection coefficient that can be achieved by any second medium. If the second medium is not a perfect reflector, for example, if the second medium is a dielectric or a plasma, certain amount of power can be transmitted into the second medium and ${\left(R_s^{\prime }\right)}^2$ will not be one. However, the power reflection coefficient ρ can be expressed for such a case as α times ${\left(R_s^{\prime }\right)}^2$

Figure 14.5 shows the plot of α vs. β. The parameter C = 0.5, that is, the angle of incidence θ = 60°. The angle of reflection θ′ as seen in ∑′ and the angle of reflection θ_r as seen in ∑ are also shown.

Several interesting points can be noted from Figure 14.5. α is zero at $\mathrm{\beta }=\frac{1-S}{C}$, since the angle of reflection θ_r, as seen in ∑, is 90° at this value of β. In the range $\frac{1-S}{C}<\mathrm{\beta }<C$, θ_r > 90° and α is thus negative.

This means that in this range of β, the reflected wave travels towards the medium 2 as seen from ∑. For an interpretation of Figure 14.5, for the range C < β < 1, see [12], which is given as Appendix 14A.

Another surprising result is that α is greater than one for the range of −1 < β < 0. This phenomenon can be explained by noting that mechanical power is supplied to the system to maintain the uniform motion of the conductor against the radiation pressure. A detailed calculation of the electric power P_el, the rate of the decrease of the stored energy in the fields P_s, and the mechanical power supplied to the medium to overcome the radiation pressure force P_m for the conductor as well as the dielectric half-space are given in Appendix 14B [23].

This appendix contains many more references on the topic.

14.10Constitutive Relations for a Moving Dielectric

Minkowsky theory is developed based on

for a simple loss-free medium, where ε′ and μ′ are the parameters measured in ∑′. Using field transformations, one can obtain the constitutive relations of the Minkowsky theory.

Equations 14.113 and 14.114 are obtained using (14.92) and (14.115):