# Boundary conditions of a plane wave on a conductor

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1. Apr 24, 2017

### 1v1Dota2RightMeow

1. The problem statement, all variables and given/known data
Consider a plane monochromatic wave incident on a flat conducting surface. The incidence angle is $θ$. The wave is polarized perpendicular to the plane of incidence. Find the radiation pressure (time-averaged force per unit area) exerted on the surface.

2. Relevant equations
Radiation pressure for reflection $\to$$P_{reflected} = \frac{2\langle S \rangle \cos ^2 (\theta_I)}{c}$

$\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$

$\vec{E} = E_0 e^{i(\vec{k} \cdot \vec{r} - wt)}\hat y$

$\vec{B} = \frac{1}{v_1}E_0 e^{i(\vec{k} \cdot \vec{r} - wt)}(-\cos \theta_I \hat x + \sin \theta_I \hat z)$

3. The attempt at a solution
$\vec{E} \times \vec{B} = \frac{1}{v_1}E_0^2 e^{2i(\vec{k} \cdot \vec{r} - wt)}(\sin \theta_I \hat x + \cos \theta \hat z)$

$\vec{k} \cdot \vec{r} = zk \sin \theta_I + xk \cos \theta_I$

$\to P = \frac{2}{\mu_0 c^2}E_0^2 e^{2i((zk \sin \theta_I + xk \cos \theta_I) - wt)}(\sin \theta_I \hat x + \cos \theta_I \hat z)$

But I know this isn't right because I need to find the time average of the Ponyting vector. How do I do so?

2. Apr 29, 2017

### PF_Help_Bot

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