Radiation pressure (Jackson exercise)

Question

Here's an exercise from Jackson:

A plane wave is incident normally on a perfectly absorbing flat screen. From the law of conservation of linear momentum show that the pressure exerted from the screen is equal to the field energy per unit volume in the wave.

My reasoning was this:

Suppose the screen is parallel to the $zy$ plane at $x=0$. For simplicity let the fields be:

$$\textbf{E}=E_0e^{i(kx-\omega t+\delta)} \hat{\textbf{y}}$$ $$\textbf{B}=\frac{E_0}{c}e^{i(kx-\omega t+\delta)} \hat{\textbf{z}}$$

The force exerted on a volume $dV$ is therefore:

$$d\textbf{f}=\textbf{n}\cdot TdS+\frac{1}{c^2}\frac{\partial \textbf{S}}{\partial t}dSdx$$ Where $T$ is the Maxwell stress tensor and $\textbf{n}$ is the normal vector to the surface.

Since the screen is perfectly flat we let $dx \rightarrow 0$ and we are left with

$$\frac{d\textbf{f}}{dS}=\textbf{n}\cdot T$$

After some calculation the result is:

$$\frac{d\textbf{f}}{dS}=\epsilon_0E_0^2\cos^2(kx-\omega t+ \delta)\hat{\textbf{x}}$$

Taking the module and mediating in time the result is there, but this left me a bit baffled. I don't see where I used the fact that the screen is perfectly absorbing! (And, honestly, I wouldn't know how to mathematically formulate such a propriety). Moreover, why is the mean in time usually taken? Do you have a more rigorous approach to this?

Answer 1

You assume that the screen is perfectly absorbing when you use this field. By uses this field you are assuming that the electromagnetic field pass through the material without reflection. Furthermore, you are assuming that you don't have field in the another side of the surface. $$ \textbf{E}=E_0e^{i(kx-\omega t+\delta)} \hat{\textbf{y}}, \ \ x<0 $$ $$ \textbf{B}=\frac{E_0}{c}e^{i(kx-\omega t+\delta)} \hat{\textbf{z}}, \ \ x<0 $$ $$ \textbf{E}=0 \ \ and \ \ \textbf{B}=0, \ \ when \ \ x>0 $$ For example, if your screen is perfectly reflective, then you need to use the superposition of the incoming field and the outgoing field (reflected), getting twice of your result. If your screen is perfectly transparent, then you need to use the field in the two side of the surface, getting zero.

You can deduce from Maxwell equation plus boundary conditions, the full electromagnetic field. This full electromagnetic field tells you if you have or not a reflection and so forth. For example, if your screen is made by a perfect conductor, then you get a perfect reflection. Or you can design a dispersive material with natural frequencies matching the incidents ones for your problem of totally absorptive (resonance).

You are interest in the pressure of this wave, the force in the screen oscillates very fast. Then you define the pressure in this way.

Answer 2

Instead of complex EM field energy momentum analysis an easy route to answer uses Einstein relativity E^2 = p^2c^2 + m^2c^4 and the fact that EM is automatically relativistic theory.

For an absorbing metre square of plate in one second E=c.U quantity of energy is absorbed as c times 1 sq m will be the volume of wave that hits per second, where U is the EM wave energy density.

Einstein’s relativity gives E= pc (where (small) p is momentum) for a particle or “something” moving at the speed of light needing m=0. So the momentum absorbed per second is trivial and also the pressure as force per unit area with force =dp/dt leads directly to the result desired.