A cylinder of radius r and length l is placed in an uniform electric field parallel to the axis of the cylinder. The total flux for the surface of the cylinder is given by-

Options

A. zero
B. `pi r^2`
C. `E pi r^2`
D. `2 E pi r^2`

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 Concept of Electric Flux
Electric flux (Φ) through a surface is defined as the product of the electric field (E) and the area (A) through which the field lines pass, taking into account the angle (θ) between the electric field and the normal to the surface. Mathematically, it is given by:
\[ \Phi = E \cdot A \cdot \cos(\theta) \]

### Step 2: Identify the Geometry of the Cylinder
For a cylinder of radius \( r \) and length \( l \), the surface area consists of two circular bases and a curved surface. However, since the electric field is parallel to the axis of the cylinder, we need to consider the contribution of the curved surface and the bases separately.

### Step 3: Calculate the Area of the Curved Surface
The area of the curved surface of the cylinder is given by:
\[ A_{\text{curved}} = 2 \pi r l \]

### Step 4: Determine the Electric Flux through the Curved Surface
Since the electric field is parallel to the axis of the cylinder, the angle \( \theta \) between the electric field and the normal to the curved surface is \( 0^\circ \) (cosine of 0 is 1). Therefore, the electric flux through the curved surface is:
\[ \Phi_{\text{curved}} = E \cdot A_{\text{curved}} \cdot \cos(0) = E \cdot (2 \pi r l) \cdot 1 = 2 \pi r l E \]

### Step 5: Calculate the Electric Flux through the Circular Bases
For the circular bases, the electric field is perpendicular to the normal of the bases, which means \( \theta = 90^\circ \) (cosine of 90 is 0). Thus, the electric flux through each base is:
\[ \Phi_{\text{base}} = E \cdot A_{\text{base}} \cdot \cos(90) = E \cdot (\pi r^2) \cdot 0 = 0 \]
Since there are two bases, the total flux through the bases is also zero.

### Step 6: Calculate the Total Electric Flux
The total electric flux through the entire surface of the cylinder is the sum of the flux through the curved surface and the bases:
\[ \Phi_{\text{total}} = \Phi_{\text{curved}} + \Phi_{\text{bases}} = 2 \pi r l E + 0 = 2 \pi r l E \]

### Final Answer
The total electric flux for the surface of the cylinder is given by:
\[ \Phi = 2 \pi r l E \]

---