Problem 36 - Master Resource Book in JEE Main Physics

Question

When a point source of light is \(1 \mathrm{~m}\) away from a photoelectric cell, the photoelectric current is found to be \(I \mathrm{~mA}\). If the same source is placed at \(4 \mathrm{~m}\) from the same photoelectric cells, the photoelectric current (in \(\mathrm{mA}\) ) will be (a) \(\frac{1}{16}\) (b) \(\frac{I}{4}\) (c) \(4 I\) (d) \(16 I\)

Step-by-Step Solution

Answer

The photoelectric current will be \( \frac{I}{16} \text{ mA} \), which is option (a).

Step 1: Understand the Relation

The photoelectric current is directly proportional to the intensity of the light that falls on the cell, which in turn is inversely proportional to the square of the distance from the light source. This is known from the inverse square law of light.

Step 2: Define the Current Equation

Since current is proportional to intensity and intensity (I) is proportional to \( \frac{1}{d^2} \), where \(d\) is the distance, the relation is \( I_d \propto \frac{1}{d^2} \).

Step 3: Calculate Initial Intensity

When the source is at \(1 \text{ m}\), the intensity can be expressed as \( I_1 \propto \frac{1}{1^2} = 1 \). Hence, the current is \( I \text{ mA} \).

Step 4: Calculate New Intensity

For the new distance, \(4 \text{ m}\), the intensity becomes \( I_4 \propto \frac{1}{4^2} = \frac{1}{16} \).

Step 5: Find the New Current Value

Using the relation \( I_4 \text{ mA} \), the current at \(4 \text{ m}\) becomes \( \frac{1}{16} \cdot I \text{ mA} \).

Step 6: Match with Options

Translate the result to the closest option: the value \( \frac{I}{16} \text{ mA} \) matches option (a), which is \( \frac{1}{16} \).

Key Concepts

Understanding the Inverse Square Law

The inverse square law is a crucial principle when discussing light and its intensity. It states that the intensity of light or any other form of radiation from a point source decreases with the square of the distance from the source. This means that as you move away from a light source, the intensity of light diminishes rapidly.
For instance, if you double the distance from a light source, the intensity becomes a quarter (1/4) of what it was. Similarly, if the distance is tripled, the intensity drops to one-ninth (1/9).
This law is represented mathematically as:

• \[ I \propto \frac{1}{d^2} \]

Here, \( I \) represents the intensity, and \( d \) is the distance from the source. So, it's clear that even a small increase in distance significantly lessens the intensity of light reaching a defined point.

Unraveling the Intensity of Light

The concept of light intensity is pivotal in understanding how light interacts with different surfaces. Intensity refers to the amount of energy that a light wave carries per unit area. Simply put, it's how bright the light appears to be from a certain distance.
When talking about a photoelectric cell, intensity is directly related to the amount of photoelectric current generated. More intense light increases the energy available, and thus the current generated.
The connection between distance and intensity is intertwined with the inverse square law; as the distance increases, intensity decreases, and vice versa. This interaction plays a critical role when calculating changes in photoelectric current as the position of the light source shifts.
It's key to remember that high intensity equates to greater photon interaction with the surface of the photoelectric cell, which results in more substantial currents.

Exploring Photoelectric Current

Photoelectric current arises when photons from a light source interact with a material, typically a metal, and cause the release of electrons. This release forms the photoelectric effect, a fundamental principle for devices like solar panels and photoelectric cells.
The magnitude of the photoelectric current depends on the intensity of the light. Stronger light, with a high intensity, results in a greater number of emitted electrons, thus a higher current. Conversely, weaker light leads to a reduced current due to fewer electrons being released.
In the context of the photoelectric cells, understanding how current behaves with changing distances involves recognizing that distance affects light intensity through the inverse square law. As shown in our example, moving a light source from 1 m to 4 m resulted in the intensity decreasing by a factor of 16, leading to the current also reducing significantly by the same factor (since current is directly proportional to intensity).
This deepens our understanding of how both light and distance impact electrical values in practical applications.