The electric field in a certain region is given by `E=5 hat(i)-3hat(j) kv//m`. The potential difference `V_(B)-V_(A)` between points a and B having coordinates (4, 0, 3) m and (10, 3, 0) m respectively, is equal to

Options

A. 21 kV
B. `-21 kV`
C. `39 kV`
D. `-39 kV`

Answer

To solve the problem, we need to find the potential difference \( V_B - V_A \) between points A and B given the electric field \( \mathbf{E} = 5 \hat{i} - 3 \hat{j} \, \text{kV/m} \) and the coordinates of points A and B.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A and B:**
- Point A has coordinates \( (4, 0, 3) \) m.
- Point B has coordinates \( (10, 3, 0) \) m.

2. **Calculate the Displacement Vector \( \mathbf{r} \):**
\[ \mathbf{r} = \mathbf{B} - \mathbf{A} = (10 - 4) \hat{i} + (3 - 0) \hat{j} + (0 - 3) \hat{k} \]
\[ \mathbf{r} = 6 \hat{i} + 3 \hat{j} - 3 \hat{k} \]

3. **Use the Relation for Potential Difference:**
The potential difference between two points in an electric field is given by:
\[ V_B - V_A = - \mathbf{E} \cdot \mathbf{r} \]

4. **Calculate the Dot Product \( \mathbf{E} \cdot \mathbf{r} \):**
\[ \mathbf{E} = 5 \hat{i} - 3 \hat{j} \, \text{kV/m} \]
\[ \mathbf{E} \cdot \mathbf{r} = (5 \hat{i} - 3 \hat{j}) \cdot (6 \hat{i} + 3 \hat{j} - 3 \hat{k}) \]
\[ = 5 \cdot 6 + (-3) \cdot 3 + 0 \]
\[ = 30 - 9 = 21 \, \text{kV} \]

5. **Calculate the Potential Difference:**
\[ V_B - V_A = - (21 \, \text{kV}) = -21 \, \text{kV} \]

### Final Answer:
The potential difference \( V_B - V_A \) is \( -21 \, \text{kV} \).