A `5 W` source emits monochromatic light of wavelength `5000 Å`. When placed `0.5 m` away , it liberates photoelectrons from a photosensitive metallic surface . When the source is moved to a distance of `1.0 m` the number of photoelectrons liberated will be reduced by a factor of

Options

A. 8
B. 16
C. 2
D. 4

Answer

To solve the problem step-by-step, we will analyze the relationship between the power of the light source, the distance from the source, and the number of photoelectrons liberated.

### Step 1: Understand the Relationship Between Power and Distance
The power \( P \) of a light source is distributed over the surface area of a sphere as the distance \( R \) from the source increases. The intensity \( I \) of light at a distance \( R \) can be expressed as:
\[ I = \frac{P_0}{4 \pi R^2} \]
where \( P_0 \) is the power of the source.

### Step 2: Relate Intensity to Number of Photoelectrons
The number of photoelectrons \( n \) liberated is directly proportional to the intensity of light:
\[ n \propto I \]
Thus, we can write:
\[ n \propto \frac{P_0}{4 \pi R^2} \]
This implies that the number of photoelectrons is inversely proportional to the square of the distance:
\[ n \propto \frac{1}{R^2} \]

### Step 3: Set Up the Ratio of Photoelectrons at Two Distances
Let \( n_1 \) be the number of photoelectrons at distance \( R_1 = 0.5 \, m \) and \( n_2 \) be the number of photoelectrons at distance \( R_2 = 1.0 \, m \). We can express the ratio of the number of photoelectrons as:
\[ \frac{n_1}{n_2} = \frac{R_2^2}{R_1^2} \]

### Step 4: Substitute the Distances
Substituting the values of \( R_1 \) and \( R_2 \):
\[ \frac{n_1}{n_2} = \frac{(1.0)^2}{(0.5)^2} = \frac{1}{0.25} = 4 \]

### Step 5: Find the Reduction Factor
From the above ratio, we can find \( n_2 \) in terms of \( n_1 \):
\[ \frac{n_2}{n_1} = \frac{1}{4} \]
This indicates that the number of photoelectrons \( n_2 \) is one-fourth of \( n_1 \).

### Conclusion
The number of photoelectrons liberated when the source is moved to a distance of \( 1.0 \, m \) is reduced by a factor of \( 4 \).