A cylinder of radius `R` and length `L` is placed in the uniform electric field `E` parallel to the cylinder axis.The total flux from the two flat surface of the cylinder is given by

Options

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

Answer

To find the total electric flux from the two flat surfaces of a cylinder placed in a uniform electric field parallel to its axis, we can follow these steps:

### Step 1: Understand the Setup
We have a cylinder with radius \( R \) and length \( L \). The electric field \( E \) is uniform and parallel to the axis of the cylinder. The two flat surfaces of the cylinder are the circular ends.

### Step 2: Identify the Area Vectors
The area vector \( \vec{A} \) for the flat surfaces (circular ends) of the cylinder points outward from the surface. For the top flat surface, the area vector points in the direction of the electric field, while for the bottom flat surface, it points in the opposite direction.

### Step 3: Calculate the Electric Flux through Each Flat Surface
The electric flux \( \Phi \) through a surface is given by the formula:
\[ \Phi = \vec{E} \cdot \vec{A} = E \cdot A \cdot \cos(\theta) \]
where \( \theta \) is the angle between the electric field and the area vector.

- For the top flat surface:
- Area \( A = \pi R^2 \)
- Angle \( \theta = 0^\circ \) (since the area vector is in the same direction as the electric field)
- Thus, the flux through the top surface \( \Phi_1 \) is:
\[ \Phi_1 = E \cdot \pi R^2 \cdot \cos(0) = E \cdot \pi R^2 \]

- For the bottom flat surface:
- Area \( A = \pi R^2 \)
- Angle \( \theta = 180^\circ \) (since the area vector is in the opposite direction to the electric field)
- Thus, the flux through the bottom surface \( \Phi_2 \) is:
\[ \Phi_2 = E \cdot \pi R^2 \cdot \cos(180) = -E \cdot \pi R^2 \]

### Step 4: Calculate the Total Flux
The total electric flux \( \Phi \) through the two flat surfaces of the cylinder is the sum of the flux through each surface:
\[ \Phi = \Phi_1 + \Phi_2 = E \cdot \pi R^2 + (-E \cdot \pi R^2) = 0 \]

### Conclusion
The total electric flux from the two flat surfaces of the cylinder is zero:
\[ \Phi = 0 \]