Boundary conditions of a plane wave on a conductor

[1v1Dota2RightMeow]

Homework Statement

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.

Homework Equations

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

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

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

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

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?

 

[mark726]

To find the time average of the Poynting vector, you can use the following equation:

$\langle \vec{S} \rangle = \frac{1}{T} \int_{0}^{T} \vec{S} dt$

Where T is the period of the wave. This will give you the average value of the Poynting vector over one full cycle of the wave.

Substituting your expressions for $\vec{E}$ and $\vec{B}$ into the equation for $\vec{S}$, you can then use the above equation to find the time average of the Poynting vector. From there, you can use the equation for the radiation pressure to find the time-averaged force per unit area exerted on the surface.