fI the carbon diode laser travels is aconducting medium with per permiability
My and premitivity &1,
(1) Derive an expression for the wave equation in a conducting medium
using Amphere-Maxwell equation. (6 m a r k s ) What si the form of a wave vector of a field in a conducting medium? (4 marks)
(inn)
Write down the form of a solution for the wave equation using the wave vector define in (c)-(ii). (4 m a r k s )


### (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

If the carbon diode laser travels in a conducting medium with permeability μ and permittivity ε₁, (i) Derive an expression for the wave equation in a conducting medium using Ampère-Maxwell equation. (6 marks) What is the form of the wave vector of a field in a conducting medium? (4 marks) (iii) Write down the form of a solution for the wave equation using the wave vector defined in (ii). (4 marks)

Learn how to derive the wave equation in a conducting medium using Ampère-Maxwell law. This problem explores key electromagnetic theory concepts such as the wave vector, Ohm’s law, and Faraday’s law. Suitable for undergraduate physics and electrical engineering students.

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