(a)  Write down Maxwell’s equations for a conducting medium of conductivity s.

(b)  A plane wave of low frequency is propagating in the z-direction inside the conducting medium. Let where E₀, B₀, and k are complex. Use Maxwell’s equations to show that k= k₁+ik₂, Calculate the ratio of the amplitude of the two fields, and their phase difference.

(c)  Calculate the energy flux (time averaged Poynting vector).

,

and the constitutive relations for a linear, isotropic medium

.

Let J=sE + J_b, where s is the conductivity of the medium and J_b represents forced currents, such as battery currents.

(a)  What is the relationship between the field quantities E and B and the potentials A and V?
(b)  E and B do not specify A and V uniquely. Let .  Show that Maxwell's equations are satisfied by A' and V' if they are satisfied by A and V.
(c)  Briefly discuss the gauge transformation represented in part b and its implications.
(d)  Write the relationship between the fields given in answers to part a in terms of H, D, A, and V.
(e)  Find the wave equation for A in terms of V, m , e , J_b, s , and  .
(f)  Starting with Maxwell's equations, derive the wave equation for V in the Lorentz gauge.