Question

A point source is placed at origin. Its intensity at distance of 2 cm from source is I then intensity at distance 4 cm from the source shall be.

• A. \(\frac{I}{2}\)
• B. \(\frac{I}{16}\)
• C. \(\frac{I}{4}\)
• D. I

Answer 1

The Correct Option is C

Determine the light intensity from a point source at 4 cm, given that its intensity is I at 2 cm.

Concept Utilized:

Point source light intensity adheres to the inverse square law. This principle dictates that intensity (E) is inversely proportional to the square of the distance (d) from the source. This relationship arises because a point source emits energy uniformly in all directions, and as distance increases, this energy disperses over the surface of progressively larger spheres.

The surface area of a sphere with radius \(d\) is \(A = 4\pi d^2\). For a source with constant power P, the intensity (Power per unit area) is defined as:

\[ E = \frac{P}{A} = \frac{P}{4\pi d^2} \]

This equation reveals the following proportionality:

\[ E \propto \frac{1}{d^2} \]

Solution Steps:

Step 1: Establish the relationship between intensities at two distinct distances.

Let \(E_1\) represent the intensity at distance \(d_1\), and \(E_2\) represent the intensity at distance \(d_2\). Applying the inverse square law, the ratio is expressed as:

\[ \frac{E_2}{E_1} = \frac{1/d_2^2}{1/d_1^2} = \frac{d_1^2}{d_2^2} = \left(\frac{d_1}{d_2}\right)^2 \]

Step 2: Identify the given parameters from the problem statement.

The initial intensity is \(E_1 = I\).

The initial distance is \(d_1 = 2 \, \text{cm}\).

The final distance is \(d_2 = 4 \, \text{cm}\).

The objective is to determine the final intensity, \(E_2\).

Step 3: Substitute the known values into the derived ratio equation.

\[ \frac{E_2}{I} = \left(\frac{2 \, \text{cm}}{4 \, \text{cm}}\right)^2 \]

Step 4: Compute the final intensity \(E_2\).

First, simplify the fraction within the parentheses:

\[ \frac{E_2}{I} = \left(\frac{1}{2}\right)^2 \]

Next, square the simplified fraction:

\[ \frac{E_2}{I} = \frac{1}{4} \]

Finally, isolate \(E_2\) to find its value:

\[ E_2 = \frac{I}{4} \]

The intensity at a distance of 4 cm from the source is calculated to be I/4.