Electric field in a region is uniform and is given by `vec(E)-a hati+b hatj+c hatk`. Electric flux associated with a surface of area `vec(A)=pi R^(2)hati` is

Options

A. `a pi R^(2)`
B. `3a pi R^(2)`
C. `2ab R`
D. `acR`

Answer

To find the electric flux associated with a surface area given by \(\vec{A} = \pi R^2 \hat{i}\) in a uniform electric field \(\vec{E} = a \hat{i} + b \hat{j} + c \hat{k}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electric Field and Area Vector**:
- The electric field is given as \(\vec{E} = a \hat{i} + b \hat{j} + c \hat{k}\).
- The area vector is given as \(\vec{A} = \pi R^2 \hat{i}\).

2. **Understand Electric Flux**:
- Electric flux (\(\Phi\)) through a surface is defined as the dot product of the electric field vector and the area vector:
\[ \Phi = \vec{E} \cdot \vec{A} \]

3. **Calculate the Dot Product**:
- The dot product \(\vec{E} \cdot \vec{A}\) can be calculated as follows:
\[ \Phi = (a \hat{i} + b \hat{j} + c \hat{k}) \cdot (\pi R^2 \hat{i}) \]
- Using the properties of the dot product, we can expand this:
\[ \Phi = a \hat{i} \cdot (\pi R^2 \hat{i}) + b \hat{j} \cdot (\pi R^2 \hat{i}) + c \hat{k} \cdot (\pi R^2 \hat{i}) \]

4. **Evaluate Each Term**:
- The dot product of \(\hat{i} \cdot \hat{i} = 1\), while \(\hat{j} \cdot \hat{i} = 0\) and \(\hat{k} \cdot \hat{i} = 0\):
\[ \Phi = a (\pi R^2) + b(0) + c(0) \]
- Thus, we have:
\[ \Phi = a \pi R^2 \]

5. **Final Result**:
- The electric flux associated with the surface area is:
\[ \Phi = a \pi R^2 \]