Animated Solution for Physics - Dual Nature of Matter and Radiation: A photocell is illuminated by a small bright source placed $1\text{ m}$ away. When the same source of light is placed $1/2\text{ m}$ away, the number of electrons emitted by photocathode would

Catching Photons

Imagine you are standing in a dark room with a single, incredibly bright light bulb in the center. This bulb is our point source, and it is throwing out photons in every single direction, like a cosmic sprinkler system.

Now, we place a photocell—a device designed to catch these photons and release electrons—at a distance of $1\mathrm{m}$ from the bulb.

The question asks us a very practical thing: what happens to the number of electrons emitted if we move this photocell closer, to a distance of $0.5\mathrm{m}$? To answer this, we need to understand how light travels through space.

The Inverse Square Law

As photons travel outward from a point source, they don't just move in a straight line; they spread out spherically. Imagine blowing up a balloon. As the balloon gets larger, the rubber stretches, and the color becomes less intense because the same amount of rubber is covering a much larger area.

Light behaves in the exact same way. The area of a sphere is given by $4\pi r^2$. Because the total power $P$ emitted by the bulb is constant, the power per unit area—which we call intensity $I$—must decrease as the sphere gets larger.

Mathematically, this is expressed as:

This tells us a profound truth about our universe: the intensity of light from a point source is inversely proportional to the square of the distance.

If you double the distance, the light doesn't just become half as bright; it becomes one-fourth as bright!

Now, let's connect this geometric reality to the quantum world of the photoelectric effect. When light hits our photocell, it's not just a continuous wave; it's a stream of discrete packets of energy called photons.

According to the laws of the photoelectric effect, assuming the frequency of light is above the threshold, every photon has a chance to knock out an electron. Therefore, the number of electrons emitted per second, which we will call $n_e$, is directly proportional to the number of photons hitting the surface per second.

And what determines the number of photons hitting the surface? The intensity of the light!

Combining this with our inverse square law, we arrive at our master relationship for this problem:

We are now ready to solve the problem. We have two scenarios. In Case 1, the distance $r_1$ is $1\mathrm{m}$, and the emission rate is $n_1$. In Case 2, the distance $r_2$ is $0.5\mathrm{m}$, and the emission rate is $n_2$.

Because $n_e$ is inversely proportional to $r^2$, we can set up a ratio.

This is where you must be careful not to make a silly mistake. Notice how $n_2$ is in the numerator on the left, but $r_2$ is in the denominator on the right. This is the mathematical signature of an inverse relationship.

Let's substitute our given values into the equation:

Since $1$ divided by $0.5$ is exactly $2$, the equation simplifies beautifully:

By moving the photocell twice as close, we didn't just double the number of electrons; we quadrupled them! The number of electrons emitted increases by a factor of 4.

Before we wrap up, let's do a quick thought experiment. What if our light source wasn't a small bulb, but a very long fluorescent tube?

A long tube acts as a line source, not a point source. Instead of spreading out spherically, the light spreads out cylindrically. The surface area of a cylinder is $2\pi rL$, which means the intensity would fall off as $1/r$, not $1/r^2$.

If we had used a line source in this problem, halving the distance would have only doubled the number of electrons. Always pay attention to the geometry of your source!