The electric field in a certain region is given by `E=(5hati-3hatj)kV//m`. Find the difference in potential `V_B-V_A`. If A is at the origin and point B is at a. (0,0,5)m, b. (4,0,3) m.`

Explanation

To find the difference in potential \( V_B - V_A \) in the given electric field \( \mathbf{E} = (5 \hat{i} - 3 \hat{j}) \, \text{kV/m} \), we will use the relationship between electric field and potential difference:

\[ V_B - V_A = - \int_A^B \mathbf{E} \cdot d\mathbf{r} \]

where \( d\mathbf{r} \) is the displacement vector from point A to point B.

### Step 1: Identify the points A and B
- Point A is at the origin: \( (0, 0, 0) \)
- For part (a), point B is at \( (0, 0, 5) \)
- For part (b), point B is at \( (4, 0, 3) \)

### Step 2: Calculate the displacement vector \( \mathbf{r}_{BA} \)
- For part (a):
\[ \mathbf{r}_{BA} = \mathbf{r}_B - \mathbf{r}_A = (0, 0, 5) - (0, 0, 0) = 0 \hat{i} + 0 \hat{j} + 5 \hat{k} = 5 \hat{k} \]

- For part (b):
\[ \mathbf{r}_{BA} = \mathbf{r}_B - \mathbf{r}_A = (4, 0, 3) - (0, 0, 0) = 4 \hat{i} + 0 \hat{j} + 3 \hat{k} \]

### Step 3: Calculate the dot product \( \mathbf{E} \cdot \mathbf{r}_{BA} \)
- The electric field vector is \( \mathbf{E} = (5 \hat{i} - 3 \hat{j}) \, \text{kV/m} \).

- For part (a):
\[ \mathbf{E} \cdot \mathbf{r}_{BA} = (5 \hat{i} - 3 \hat{j}) \cdot (0 \hat{i} + 0 \hat{j} + 5 \hat{k}) = 0 \]
Thus,
\[ V_B - V_A = -\int_A^B \mathbf{E} \cdot d\mathbf{r} = 0 \]

- For part (b):
\[ \mathbf{E} \cdot \mathbf{r}_{BA} = (5 \hat{i} - 3 \hat{j}) \cdot (4 \hat{i} + 0 \hat{j} + 3 \hat{k}) = 5 \cdot 4 + (-3) \cdot 0 = 20 \]
Thus,
\[ V_B - V_A = -\int_A^B \mathbf{E} \cdot d\mathbf{r} = -20 \, \text{V} \]

### Final Results:
- For part (a): \( V_B - V_A = 0 \, \text{V} \)
- For part (b): \( V_B - V_A = -20 \, \text{V} \)