A perfectly reflecting solid hemisphere of radius R is placed in the path of a parallel beam of light of large aperture. If the beam carrries an intensity l, find the force exerted by the beam on the hemisphere.

Options

A. `(2piR^2l)/c`
B. `(piR^2l)/c`
C. `(4piR^2l)/c`
D. `(piR^2l)/(2c)`

Answer

To find the force exerted by a parallel beam of light on a perfectly reflecting solid hemisphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a perfectly reflecting solid hemisphere of radius \( R \) placed in the path of a parallel beam of light with intensity \( I \).
- The beam of light is large enough to cover the hemisphere.

2. **Calculating the Affected Area**:
- The area of the flat circular face of the hemisphere that is exposed to the light is given by:
\[ A = \pi R^2 \]

3. **Understanding Light Pressure**:
- When light strikes a surface, it exerts pressure. For a perfectly reflecting surface, the pressure \( P \) exerted by the light can be expressed as:
\[ P = \frac{I}{c} \]
- Here, \( I \) is the intensity of the light, and \( c \) is the speed of light.

4. **Calculating the Force**:
- The force \( F \) exerted by the light on the hemisphere can be calculated using the relationship between pressure and force:
\[ F = P \cdot A \]
- Substituting the expressions for pressure and area, we get:
\[ F = \left(\frac{I}{c}\right) \cdot (\pi R^2) \]
- Therefore, the force exerted by the beam on the hemisphere is:
\[ F = \frac{\pi R^2 I}{c} \]

### Final Answer:
The force exerted by the beam on the hemisphere is:
\[ F = \frac{\pi R^2 I}{c} \]