# Homework Help: How do I get this interface condition?

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1. Jan 10, 2018

### svletana

1. The problem statement, all variables and given/known data
I have an infinitely long cylinder of a dielectric material, surrounded by another dielectric material and coated with graphene which has surface conductivity $\sigma$, implying it has a superficial current. The sheet of graphene is very thin, and the dielectrics are asumed to be isotropic. There's an incident p-polarised monochromatic plane wave in the x direction, and I have to show the interface conditions are:
$$\frac{1}{\epsilon_1} \frac{dF_1}{dr} = \frac{1}{\epsilon_2} \frac{dF_2}{dr}$$

$$F_2 - F_1 = \frac{4i \pi}{\omega \epsilon_1} \sigma \frac{dF_1}{dr}$$

where F is the H field in the z direction, the subindex 1 indicates the material inside the cylinder and 2 outside the cylinder. I'm having trouble with the second one.

2. Relevant equations
The boundary conditions given is that the tangential component of E (tangential to the normal vector of the surface of the cylinder) is continuous, and that the tangential component of H is proportional to the surface conductivity density.

I have tried using Maxwell equations: $$\nabla \times H = \frac{4 \pi}{c}J - \frac{i \omega}{c} \epsilon E$$

3. The attempt at a solution
The $\phi$ component is: $$-\frac{dH_z}{dr}= \frac{4 \pi}{c}J_{\phi} - \frac{i \omega \epsilon}{c} E_{\phi}$$

If I do the integral between R- and R+ (R is the radius of the cylinder) then the left side would look like $F_2 - F_1$, which is what I'm looking for, but what about the right side? I feel like I have to use the boundary conditions given, but I'm not sure how. Please help! Thank you!

2. Jan 16, 2018

### PF_Help_Bot

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