Deriving Wave Equation - Electric Field Inside Metal

In summary, the conversation discusses deriving the wave equation for an electromagnetic wave hitting a metallic surface with conductivity σ at normal incidence. Using Ohm's law and vector calculus, the Maxwell equations are manipulated to obtain the wave equations for the electric and magnetic fields. The conversation also mentions using the separation of variables method and Fourier series to solve the wave equation for the electric field.
  • #1
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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!
 
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  • #2
Hi I solved past year in the course of partial differential equations, I rembember tha we use the separation of variables methods, I will search it, and if a I get I will tell you.
 
  • #3
Sorry, I forget to say that the way we solved was a Fourier Series, (boundary conditions included)
 

FAQ: Deriving Wave Equation - Electric Field Inside Metal

1. What is the wave equation for the electric field inside a metal?

The wave equation for the electric field inside a metal is given by E = cB, where E is the electric field, c is the speed of light, and B is the magnetic field.

2. How is the wave equation derived for the electric field inside a metal?

The wave equation for the electric field inside a metal can be derived from Maxwell's equations, specifically the equation for Faraday's law of induction and the equation for Ampere's law. By combining these two equations, the wave equation for the electric field can be obtained.

3. What is the significance of the wave equation for the electric field inside a metal?

The wave equation for the electric field inside a metal is significant because it describes the behavior of electromagnetic waves inside a metal. This equation helps us understand how the electric and magnetic fields are related and how they propagate through a conductive material.

4. What are the assumptions made when deriving the wave equation for the electric field inside a metal?

When deriving the wave equation for the electric field inside a metal, some assumptions are made. These include:

  • The material is homogeneous and isotropic (uniform and the same in all directions).
  • The material is linear (the relationship between the electric and magnetic fields is constant).
  • The material is non-magnetic (the magnetic permeability is equal to that of free space).
  • The material is non-dispersive (the speed of light is constant).

5. How does the wave equation for the electric field inside a metal differ from that for free space?

The wave equation for the electric field inside a metal differs from that for free space in that it includes the factor of the speed of light c. This is because the propagation of electromagnetic waves in a metal is affected by the material properties, such as the conductivity and permittivity, which are not present in free space.

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