Animated Solution for Physics - Electromagnetic Waves: Light is incident normally on a completely absorbing surface with an energy flux of $25{\text{ W cm}}^{-2}$. If the surface has an area of $25{\text{ cm}}^2$, the momentum transferred to the surface in $40\text{ min}$ time duration will be

Calculating Momentum Transfer

Have you ever wondered how a solar sail works in space? Even though light has no mass, it carries momentum. When photons strike a surface, they exert a tiny but measurable push. This problem is a classic demonstration of how we can calculate that exact push—the momentum transferred by an electromagnetic wave.

Imagine a completely absorbing surface. Light waves are continuously bombarding it, delivering energy and momentum. We are given three crucial pieces of information:

1. The energy flux (or intensity) of the light is $I=25{\text{ W cm}}^{-2}$. 2. The area of the surface is $A=25{\text{ cm}}^2$. 3. The time duration of the exposure is $\Delta t=40\text{ min}$.

The key word here is absorbing. When a surface completely absorbs light, the photons transfer all of their momentum to it. If the surface were perfectly reflecting, the photons would bounce back, transferring twice the momentum!

To find the momentum transferred ($\Delta p$), we use the fundamental relationship between the energy of an electromagnetic wave and its momentum:

Here, $U$ is the total energy absorbed, and $c$ is the speed of light ($3\times 10^8\text{ m/s}$). But how do we find $U$? We know that intensity ($I$) is power per unit area, and power is energy per unit time. Therefore, the total energy is the product of intensity, area, and time:

Substituting this back into our momentum equation gives us our master formula:

Let's plug in our values. Notice a beautiful trick here: the intensity is given in ${\text{W cm}}^{-2}$ and the area in ${\text{cm}}^2$. When we multiply them, the ${\text{cm}}^2$ units perfectly cancel out, saving us from tedious unit conversions!

First, let's convert the time into standard SI units (seconds):

Now, let's calculate the total energy $U$:

Finally, we divide this total energy by the speed of light to find the momentum transferred:

Rewriting this in standard scientific notation, we get our final answer:

This tiny push is the very principle that could one day propel spacecraft across the galaxy using nothing but the light of the stars!