Plane Wave in Metal

Suppose that a plane electromagnetic wave of frequency ω/2π and amplitude E₀ is normally incident on the flat surface of a semi-infinite metal of conductivity σ. Assume the frequency is low so that the displacement current inside the metal can be neglected. The magnetic permeability of the metal μ = 1.

a) Using Maxwell’s equations, derive expressions for the components of the electric and magnetic fields inside the conductor which are parallel to the surface. What is the characteristic penetration depth of the field?

b) What is the ratio of the magnetic field amplitude to the electric field amplitude inside the metal?

c) What is the power per unit area transmitted into the metal?

SOLUTION

a) Starting with Maxwell’s equations, we may follow a standard procedure to arrive at the wave equation for the fields and then the dispersion relations:

(1)

(2)

(3)

(4)

First take the curl of (4)

(5)

Using the identity

in (5), we obtain

(6)

 Figure 1.1

Inside the conductor, we must use the relations J = σE and D = εE in (6). Therefore (6) becomes

(7)

Using (2) and B = μH in (7), we obtain the wave equation for H of the wave propagating in the z direction (see Figure 1.1):

(8)

Disregarding the displacement current (which is equivalent to the condition σ >> ω), we obtain from (8)

(9)

Substituting the plane wave solution into (9) results in

(10)

or

(11)

For our case μ = 1, so we have

(12)

Assuming that the electric field in the incident wave is polarized in the x direction and the magnetic field in the y direction, and the amplitude of the field outside the conductor is E₀, we can obtain the fields inside H_t, E_t

where  is the characteristic penetration depth of the field (skin depth). From the boundary conditions Hⁱⁿ₁ = H^out_t = E₀, we have

(13)

The electric field inside the conductor E_t:

(14)

Therefore

(15)

where the phase shift –π/4 comes from the factor  Equation (14) is a special case of a more general formula

where n is a unit vector in the direction of the wave propagation, and

is the surface resistance.

b) The ratio of the amplitude of the magnetic field to that of the electric field inside the metal from (13) and (15) is in this approximation

Therefore the energy of the field inside a good conductor is mostly the magnetic energy.

c) The power per unit area P transmitted into the metal is given by the flux of the Poynting vector: