An electric field is expressed as `E = 2 hati +3 hatj`. Find the potential difference `(V_(A)-V_(B))` between two points A and B whose position vectors are given by `r_(A) = hati +2hatj` and `r_(B) = hatj +3hatk`

Options

A. `-1V`
B. `1V`
C. `2V`
D. `3V`

Explanation

To find the potential difference \( V_A - V_B \) between two points A and B in an electric field \( \mathbf{E} = 2 \hat{i} + 3 \hat{j} \), we can follow these steps:

### Step 1: Identify the position vectors
The position vectors of points A and B are given as:
- \( \mathbf{r}_A = \hat{i} + 2 \hat{j} \)
- \( \mathbf{r}_B = \hat{j} + 3 \hat{k} \)

### Step 2: Determine the coordinates of points A and B
From the position vectors, we can extract the coordinates:
- Point A: \( (x_A, y_A, z_A) = (1, 2, 0) \)
- Point B: \( (x_B, y_B, z_B) = (0, 1, 3) \)

### Step 3: Calculate the change in coordinates
Now, we need to find the change in coordinates from A to B:
- Change in \( x \): \( \Delta x = x_B - x_A = 0 - 1 = -1 \)
- Change in \( y \): \( \Delta y = y_B - y_A = 1 - 2 = -1 \)
- Change in \( z \): \( \Delta z = z_B - z_A = 3 - 0 = 3 \)

### Step 4: Calculate the potential difference
The potential difference \( V_A - V_B \) in an electric field can be calculated using the formula:
\[ V_A - V_B = -\int_A^B \mathbf{E} \cdot d\mathbf{r} \]
Where \( d\mathbf{r} = (dx, dy, dz) \).

Given \( \mathbf{E} = 2 \hat{i} + 3 \hat{j} \), we can express the dot product:
\[ \mathbf{E} \cdot d\mathbf{r} = 2 dx + 3 dy \]

Now, we will integrate this from point A to point B.

### Step 5: Set up the integral
The integral can be set up as:
\[ V_A - V_B = -\left( \int_{x_A}^{x_B} 2 \, dx + \int_{y_A}^{y_B} 3 \, dy \right) \]
Substituting the limits:
\[ V_A - V_B = -\left( \int_{1}^{0} 2 \, dx + \int_{2}^{1} 3 \, dy \right) \]

### Step 6: Evaluate the integrals
1. For the \( x \) component:
\[ \int_{1}^{0} 2 \, dx = 2[x]_{1}^{0} = 2(0 - 1) = -2 \]

2. For the \( y \) component:
\[ \int_{2}^{1} 3 \, dy = 3[y]_{2}^{1} = 3(1 - 2) = -3 \]

### Step 7: Combine the results
Now, substituting back into the potential difference equation:
\[ V_A - V_B = -(-2 - 3) = 5 \]

### Final Result
Thus, the potential difference \( V_A - V_B \) is:
\[ V_A - V_B = -5 \, \text{volts} \]