A cylinder of radius R and length L is placed in a uniform electric field E parallel to the axis. The total flux for the 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 through the surface of a cylinder placed in a uniform electric field parallel to its axis, we can follow these steps:

### Step 1: Understand the Geometry
We have a cylinder with radius \( R \) and length \( L \). The electric field \( E \) is uniform and parallel to the axis of the cylinder.

### Step 2: Identify the Surfaces of the Cylinder
The cylinder has three types of surfaces:
1. The curved lateral surface.
2. The top circular surface.
3. The bottom circular surface.

### Step 3: Calculate the Flux through the Curved Surface
For the curved surface, the electric field is parallel to the axis of the cylinder. The area vector \( dA \) of the curved surface is perpendicular to the electric field. Therefore, the angle \( \theta \) between the electric field \( E \) and the area vector \( dA \) is \( 90^\circ \).

Using the formula for electric flux:
\[ \Phi = \int \vec{E} \cdot d\vec{A} \]
Since \( \cos(90^\circ) = 0 \), the flux through the curved surface is:
\[ \Phi_{\text{curved}} = E \cdot A \cdot \cos(90^\circ) = 0 \]

### Step 4: Calculate the Flux through the Top Surface
For the top surface, the electric field is parallel to the area vector. The angle \( \theta \) is \( 0^\circ \). The area of the top surface \( A \) is given by:
\[ A = \pi R^2 \]
Thus, the flux through the top surface is:
\[ \Phi_{\text{top}} = E \cdot A \cdot \cos(0^\circ) = E \cdot (\pi R^2) \cdot 1 = E \pi R^2 \]

### Step 5: Calculate the Flux through the Bottom Surface
For the bottom surface, the area vector points in the opposite direction to the electric field. The angle \( \theta \) is \( 180^\circ \). Therefore, the flux through the bottom surface is:
\[ \Phi_{\text{bottom}} = E \cdot A \cdot \cos(180^\circ) = E \cdot (\pi R^2) \cdot (-1) = -E \pi R^2 \]

### Step 6: Calculate the Total Flux
The total flux through the entire surface of the cylinder is the sum of the fluxes through all three surfaces:
\[ \Phi_{\text{total}} = \Phi_{\text{curved}} + \Phi_{\text{top}} + \Phi_{\text{bottom}} = 0 + E \pi R^2 - E \pi R^2 = 0 \]

### Conclusion
The total electric flux through the surface of the cylinder is:
\[ \Phi_{\text{total}} = 0 \]