(i) Deriving the wave equation in a conducting medium using Ampère-Maxwell law (6 marks)

The Ampère-Maxwell law in a medium with conductivity, permittivity, and permeability is given by:

$\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}$ where:

• $\mathbf{H}$ is the magnetic field,
• $\mathbf{J}$ is the current density,
• $\mathbf{D} = \epsilon \mathbf{E}$ is the electric displacement field,
• $\epsilon$ is the permittivity of the medium.

In a conducting medium, the current density is related to the electric field by Ohm's law: $\mathbf{J} = \sigma \mathbf{E}$ where $\sigma$ is the electrical conductivity.

Substitute this into the Ampère-Maxwell equation: $\nabla \times \mathbf{H} = \sigma \mathbf{E} + \epsilon \frac{\partial \mathbf{E}}{\partial t}$

Now, using Faraday's law: $\nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t}$ where $\mu$ is the permeability of the medium.

Taking the curl of Faraday’s law: [ \nabla \times (\nabla \times \mathbf{E}) = -\mu \frac