A point source of light is 60 cm from a screen and is kept at the focus of a concave mirror which reflects light on the screen. The focal length of the mirror is 20 cm. The ratio of average intensities of the illumination on the screen when the mirror is present and when the mirror is removed is:

Options

A. `36:1`
B. `37:1`
C. `49:1`
D. `10:1`

Answer

To solve the problem, we need to find the ratio of average intensities of illumination on the screen when a concave mirror is present and when it is removed.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Distance from the point source to the screen, \( d = 60 \, \text{cm} \)
- Focal length of the concave mirror, \( f = 20 \, \text{cm} \)
- Distance from the point source to the mirror, \( d_m = f = 20 \, \text{cm} \)

2. **Calculate the Distance from the Source to the Mirror:**
- The distance from the source to the mirror is already given as \( 20 \, \text{cm} \).

3. **Calculate the Area of Illumination with the Mirror Present:**
- When the mirror is present, the light from the source is reflected to the screen.
- The area \( A_1 \) over which the power is distributed is the area of a sphere with radius equal to the distance from the source to the mirror:
\[ A_1 = 4\pi (20 \, \text{cm})^2 = 4\pi \times 400 \, \text{cm}^2 = 1600\pi \, \text{cm}^2 \]

4. **Calculate the Intensity with the Mirror Present:**
- The intensity \( I_1 \) when the mirror is present can be expressed as:
\[ I_1 = \frac{P}{A_1} = \frac{P}{1600\pi} \]

5. **Calculate the Area of Illumination when the Mirror is Removed:**
- When the mirror is removed, the area \( A_2 \) is the area of a sphere with radius equal to the distance from the source to the screen:
\[ A_2 = 4\pi (60 \, \text{cm})^2 = 4\pi \times 3600 \, \text{cm}^2 = 14400\pi \, \text{cm}^2 \]

6. **Calculate the Intensity when the Mirror is Removed:**
- The intensity \( I_2 \) when the mirror is removed can be expressed as:
\[ I_2 = \frac{P}{A_2} = \frac{P}{14400\pi} \]

7. **Calculate the Ratio of Intensities:**
- The ratio of average intensities \( \frac{I_1}{I_2} \) is:
\[ \frac{I_1}{I_2} = \frac{\frac{P}{1600\pi}}{\frac{P}{14400\pi}} = \frac{14400}{1600} = 9 \]

8. **Calculate the Average Ratio of Intensities:**
- The average intensity ratio when the mirror is present and when it is removed is:
\[ \text{Average Ratio} = \frac{I_1 + I_2}{I_2} = \frac{I_1}{I_2} + 1 = 9 + 1 = 10 \]

9. **Final Result:**
- The ratio of average intensities of illumination on the screen when the mirror is present to when it is removed is:
\[ \text{Ratio} = 10:1 \]