A hemispherical body of radius R is placed in a uniform electric field E. If the field E is parallel to the base of the hemisphere the flux linked with it is

Options

A. `2piRE`
B. `2piR^(2)E`
C. `piR^(2)E`
D. zero

Answer

To find the electric flux linked with a hemispherical body of radius \( R \) placed in a uniform electric field \( E \) that is parallel to the base of the hemisphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a hemisphere with its flat base facing downwards and the curved surface facing upwards. The electric field \( E \) is directed parallel to the flat base of the hemisphere.

2. **Electric Flux Definition**:
- Electric flux \( \Phi \) through a surface is defined as:
\[ \Phi = \int \vec{E} \cdot d\vec{A} \]
- For a uniform electric field, this simplifies to:
\[ \Phi = E \cdot A \cdot \cos \theta \]
- Where \( A \) is the area of the surface and \( \theta \) is the angle between the electric field and the normal to the surface.

3. **Analyzing the Flat Base**:
- For the flat base of the hemisphere, the area vector \( d\vec{A} \) is directed perpendicular to the base (upwards), while the electric field \( \vec{E} \) is parallel to the base (horizontally).
- Therefore, the angle \( \theta \) between \( \vec{E} \) and \( d\vec{A} \) is \( 90^\circ \).

4. **Calculating Flux through the Base**:
- Since \( \cos 90^\circ = 0 \), the flux through the flat base is:
\[ \Phi_{\text{base}} = E \cdot A \cdot \cos 90^\circ = 0 \]

5. **Analyzing the Curved Surface**:
- For the curved surface of the hemisphere, the electric field lines enter and exit the surface.
- The flux entering the curved surface is equal to the flux exiting the surface. Thus, the net flux through the curved surface is also zero.

6. **Total Electric Flux**:
- Since both the flat base and the curved surface contribute zero flux, the total electric flux linked with the hemisphere is:
\[ \Phi_{\text{total}} = \Phi_{\text{base}} + \Phi_{\text{curved}} = 0 + 0 = 0 \]

### Conclusion:
The electric flux linked with the hemispherical body in the given uniform electric field is:
\[ \Phi = 0 \]