Function profile of toothed and plane parallel plates, then fourier transform.

[MadMax]

I want to write the dielectric profile of the following system, so I can then write its Fourier transform as an integral...

The plates are semi-infinite.

So far I have:

Epsilon, for: -infinity < z <= 0

1, for: 0 < z < ?

Epsilon, for: ? <= z < +infinity

I need to find the expression given by ? to complete the profile but have no idea how given that its not continious. That will do for now. Any help would be much appreciated. Thanks.

 

[Jhero]

Hello, thank you for sharing your question. I would be happy to assist you in writing the dielectric profile for this system.

First, let's define some variables to make the discussion clearer. Let's say the distance between the plates is d, and the dielectric constant of the material between the plates is ε. Additionally, let's denote the position along the z-axis as z.

Based on the information provided, we can divide the system into three regions: z < 0 (region 1), 0 < z < d (region 2), and z > d (region 3).

In region 1, the material is vacuum, so the dielectric constant is 1. In region 2, the material is the same as the one between the plates, so the dielectric constant is ε. In region 3, the material is also vacuum, so the dielectric constant is 1.

Therefore, the dielectric profile for this system can be written as:

ε(z) = 1, for z < 0

ε(z) = ε, for 0 < z < d

ε(z) = 1, for z > d

To find the expression for ? in region 2, we need to consider the boundary conditions at z = 0 and z = d. Since the plates are semi-infinite, the electric field must be continuous at these boundaries. Therefore, we can write the following equations:

E(z = 0^-) = E(z = 0^+)

E(z = d^-) = E(z = d^+)

Using the definition of the electric field as E = -∂φ/∂z, where φ is the potential, and the fact that the potential must also be continuous at the boundaries, we can write:

-∂φ/∂z (z = 0^-) = -∂φ/∂z (z = 0^+)

-∂φ/∂z (z = d^-) = -∂φ/∂z (z = d^+)

Now, we can use the Fourier transform to write the potential as an integral:

φ(z) = ∫[φ(k)e^ikz]dk

where φ(k) is the Fourier transform of the potential and k is the wavevector.

Using this, we can rewrite the above equations as:

-ikφ(k) (z =