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. be reduced by a factor of 2
B. be reduced by a factor of 4
C. be reduced by a factor of 8
D. be reduced by a factor of 16

Explanation

To solve the problem, we need to determine how the number of photoelectrons emitted from a photosensitive surface changes when the distance from a light source is altered. Here’s a step-by-step solution:

### Step 1: Understand the relationship between intensity and distance
The intensity (I) of light from a point source is inversely proportional to the square of the distance (r) from the source. This can be expressed mathematically as:
\[ I \propto \frac{P}{4\pi r^2} \]
where \( P \) is the power of the source.

### Step 2: Set up the initial and final conditions
- **Initial distance (R_initial)**: \( 0.5 \, m \)
- **Final distance (R_final)**: \( 1.0 \, m \)
- **Power of the source (P)**: \( 5 \, W \)

### Step 3: Determine the ratio of intensities
Since the intensity is inversely proportional to the square of the distance, we can write:
\[ \frac{I_1}{I_2} = \frac{R_{final}^2}{R_{initial}^2} \]

Substituting the values:
\[ \frac{I_1}{I_2} = \frac{(1.0)^2}{(0.5)^2} = \frac{1}{0.25} = 4 \]

### Step 4: Relate intensity to the number of photoelectrons
The number of photoelectrons emitted (N) is directly proportional to the intensity of the light:
\[ N \propto I \]

### Step 5: Set up the ratio of the number of photoelectrons
Using the relationship from step 3:
\[ \frac{N_1}{N_2} = \frac{I_1}{I_2} = 4 \]

### Step 6: Calculate the reduction factor
From the above ratio, we can express \( N_2 \) in terms of \( N_1 \):
\[ N_2 = \frac{N_1}{4} \]

This means that when the source is moved to a distance of 1.0 m, the number of photoelectrons liberated will be reduced by a factor of 4.

### Final Answer
The number of photoelectrons liberated will be reduced by a factor of **4**.

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