Animated Solution for Physics - Electromagnetic Waves: The electric field in a plane electromagnetic wave is given by $\mathbf{E}=200\cos \left[\left(\frac{0.5\times 10^3}{m}\right)x-\left(1.5\times 10^{11}\frac{\text{rad}}{s}\times t\right)\right]\frac{\text{V}}{\text{m}}\hat{j}$ If this wave falls normally on a perfectly reflecting surface having an area of $100{\text{ cm}}^2$. If the radiation pressure exerted by the EM wave on the surface during a 10 min exposure is $\frac{x}{10^9}$. Find the value of $x$.

We are given the equation of the electric field of a plane electromagnetic wave:

The first thing we need to do is extract the useful information from this intimidating equation. By comparing it to the standard wave equation $E=E_0\cos (kx-\omega t)$, we can immediately spot the peak electric field, $E_0$.

This $E_0$ is the key to unlocking the energy carried by the wave.

Electromagnetic waves carry energy, and the rate at which this energy crosses a unit area is called its intensity ($I$). The intensity is directly related to the peak electric field by the formula:

where ${\epsilon }_0$ is the permittivity of free space and $c$ is the speed of light. This tells us exactly how "strong" the wave is.

The Push of Light

Now, the wave hits a perfectly reflecting surface. Imagine throwing a rubber ball at a wall; it bounces back, delivering twice the momentum compared to if it had just stuck to the wall. Light does the same thing! Because the surface is perfectly reflecting, the change in momentum is doubled, leading to a radiation pressure $P$ given by:

Let's substitute our intensity formula into this pressure equation to see the magic happen:

Notice how beautifully the $c$ and the $2$ cancel out! We are left with a remarkably simple expression for the radiation pressure:

All that's left is to plug in the numbers. We know ${\epsilon }_0=8.85\times 10^{-12}\text{ F/m}$ and $E_0=200\text{ V/m}$.

We can rewrite this to match the format given in the question:

The question states the pressure is $\frac{x}{10^9}$. By simply comparing the numerators, we find our final answer:

Did you notice what we didn't use? The problem explicitly mentioned an area of $100{\text{ cm}}^2$ and an exposure time of $10\text{ min}$. Why were they there?

These are classic distractors! Radiation pressure is defined as force per unit area. It is an intrinsic property of the wave and the surface type, completely independent of the total area it hits or how long it shines. If the question had asked for the total force, we would have multiplied the pressure by the area. If it asked for the total momentum transferred, we would have multiplied the force by the time. But for pressure alone, those numbers are just noise.

Always trust your concepts, and don't let extra data push you off course!