- #1

svletana

- 21

- 1

## Homework Statement

I have an infinitely long cylinder of a dielectric material, surrounded by another dielectric material and coated with graphene which has surface conductivity [itex]\sigma[/itex], 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:

[tex]\frac{1}{\epsilon_1} \frac{dF_1}{dr} = \frac{1}{\epsilon_2} \frac{dF_2}{dr} [/tex]

[tex]F_2 - F_1 = \frac{4i \pi}{\omega \epsilon_1} \sigma \frac{dF_1}{dr}[/tex]

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.

## Homework 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: [tex]\nabla \times H = \frac{4 \pi}{c}J - \frac{i \omega}{c} \epsilon E[/tex]

## The Attempt at a Solution

The [itex]\phi[/itex] component is: [tex]-\frac{dH_z}{dr}= \frac{4 \pi}{c}J_{\phi} - \frac{i \omega \epsilon}{c} E_{\phi}[/tex]

If I do the integral between R- and R+ (R is the radius of the cylinder) then the left side would look like [itex]F_2 - F_1[/itex], 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!