- #1

H12504106

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## Homework Statement

Consider an electromagnetic wave hitting a metallic surface with conductivity σ

at normal incidence.

a) Derive the wave equation describing this situation. Hint: Use Ohm’s law, J = σE to

eliminate the current.

b) Solve the wave equation for the electric field to obtain the electric field inside the metal.

How far into the metal does the field propagate?

## Homework Equations

The Maxwell Equations in matter:

[itex]\epsilon\nabla \cdot\vec{E} = \rho_f [/itex]

[itex]\nabla \times \vec{E} = -\mu\dfrac{\partial \vec{H}}{\partial t}[/itex]

[itex]\nabla \cdot \vec{H} = 0[/itex]

[itex]\nabla \times \vec{H} = \sigma\vec{E} + \epsilon \dfrac{\partial \vec{E}}{\partial t}[/itex]

## The Attempt at a Solution

By manipulating the maxwell's equations above and using vector calculus, i can obtain the following:

[itex]\nabla^2\vec{E} = \mu\sigma\dfrac{\partial\vec{E}}{\partial t}+\mu\epsilon\dfrac{\partial^2 \vec{E}}{\partial t^2}[/itex] and

[itex]\nabla^2\vec{H} = \mu\sigma\dfrac{\partial\vec{H}}{\partial t}+\mu\epsilon\dfrac{\partial^2 \vec{H}}{\partial t^2}[/itex].

But i can't proceed on with part (b). How do i solve the wave equation for the electirc field? Is the solution to this wave equation exponential?

Thanks!