A surface has the area vector `vec(A) = (2 hat(i) + 3 hat(j)) m^(2)` . The flux of an electric field through it if the field is `vec(E) = 4 hat(i) (V)/(m)` :

Options

A. 8 V - m
B. 12 V - m
C. 20 V - m
D. zero

Explanation

To calculate the electric flux through a surface given the area vector and the electric field vector, we can use the formula for electric flux:

\[ \Phi_E = \vec{E} \cdot \vec{A} \]

Where:
- \(\Phi_E\) is the electric flux,
- \(\vec{E}\) is the electric field vector,
- \(\vec{A}\) is the area vector.

### Step 1: Identify the vectors
We have:
- Area vector: \(\vec{A} = 2 \hat{i} + 3 \hat{j} \, \text{m}^2\)
- Electric field vector: \(\vec{E} = 4 \hat{i} \, \text{V/m}\)

### Step 2: Compute the dot product
The dot product of two vectors \(\vec{E}\) and \(\vec{A}\) is calculated as follows:

\[ \vec{E} \cdot \vec{A} = (E_x A_x + E_y A_y + E_z A_z) \]

In our case:
- \(E_x = 4\), \(E_y = 0\), \(E_z = 0\)
- \(A_x = 2\), \(A_y = 3\), \(A_z = 0\)

Now, substituting the values into the dot product formula:

\[ \vec{E} \cdot \vec{A} = (4 \cdot 2) + (0 \cdot 3) + (0 \cdot 0) = 8 + 0 + 0 = 8 \]

### Step 3: Write the result
Thus, the electric flux \(\Phi_E\) through the surface is:

\[ \Phi_E = 8 \, \text{V m} \]

### Final Answer
The flux of the electric field through the surface is \(8 \, \text{V m}\).

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