An electric field is expressed as `vec E = 2 hat i + 3 hat j`. Find the potential difference `(V_A - V_B)` between two points `A` and `B` whose position vectors are given by `r_A = hat i + 2 hat j and r_B = 2 hat i + hat j + 3 hat k`.

Options

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

Explanation

To solve the problem of finding the potential difference \( V_A - V_B \) between two points \( A \) and \( B \) in an electric field, we can follow these steps:

### Step 1: Understand the Electric Field and Position Vectors
We are given the electric field:
\[ \vec{E} = 2 \hat{i} + 3 \hat{j} \]
and the position vectors:
\[ \vec{r_A} = \hat{i} + 2 \hat{j} \quad \text{and} \quad \vec{r_B} = 2 \hat{i} + \hat{j} + 3 \hat{k} \]

### Step 2: Set Up the Potential Difference Formula
The potential difference \( V_A - V_B \) can be calculated using the relation:
\[ V_B - V_A = -\int_{\vec{r_A}}^{\vec{r_B}} \vec{E} \cdot d\vec{l} \]
This means we need to integrate the electric field along the path from point \( A \) to point \( B \).

### Step 3: Define the Path of Integration
The differential path element \( d\vec{l} \) can be expressed in terms of its components:
\[ d\vec{l} = dx \hat{i} + dy \hat{j} \]

### Step 4: Set Up the Integral
We need to evaluate the integral:
\[ V_B - V_A = -\int_{\vec{r_A}}^{\vec{r_B}} (2 \hat{i} + 3 \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) \]
This simplifies to:
\[ V_B - V_A = -\int_{\vec{r_A}}^{\vec{r_B}} (2dx + 3dy) \]

### Step 5: Determine the Limits of Integration
From the position vectors:
- For point \( A \) at \( (1, 2) \) (i.e., \( x = 1, y = 2 \))
- For point \( B \) at \( (2, 1) \) (i.e., \( x = 2, y = 1 \))

### Step 6: Evaluate the Integral
We can break the integral into two parts:
1. Integrate with respect to \( x \) from \( 1 \) to \( 2 \):
\[ -\int_{1}^{2} 2 \, dx = -2[x]_{1}^{2} = -2(2 - 1) = -2 \]

2. Integrate with respect to \( y \) from \( 2 \) to \( 1 \):
\[ -\int_{2}^{1} 3 \, dy = -3[y]_{2}^{1} = -3(1 - 2) = 3 \]

### Step 7: Combine the Results
Now we combine the results of the two integrals:
\[ V_B - V_A = -2 + 3 = 1 \]

### Step 8: Finalize the Potential Difference
Thus, we have:
\[ V_A - V_B = - (V_B - V_A) = -1 \text{ Volt} \]

### Conclusion
The potential difference between points \( A \) and \( B \) is:
\[ V_A - V_B = -1 \text{ Volt} \]