An electric field is expressed as `vec(E) = 2hat(i) + 3hat(j)` . Find the potential difference `(V_(A) - V_(B))` between two point A and B whose position vectors are give by `vec(r_(A)) = hat(i) + 2j and vec(r_(B)) = 2hat(i) + hat(j) + 3hat(k)`

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, we can use the formula:

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

### Step 1: Identify the electric field and position vectors
The electric field is given as:

\[ \vec{E} = 2\hat{i} + 3\hat{j} \]

The position vectors for points A and B are:

\[ \vec{r}_A = \hat{i} + 2\hat{j} \]
\[ \vec{r}_B = 2\hat{i} + \hat{j} + 3\hat{k} \]

### Step 2: Determine the differential displacement vector \( d\vec{r} \)
To find \( d\vec{r} \), we first need to express the path from A to B. The displacement vector \( \vec{r} \) from A to B can be calculated as:

\[ d\vec{r} = \vec{r}_B - \vec{r}_A = (2\hat{i} + \hat{j} + 3\hat{k}) - (\hat{i} + 2\hat{j}) = (2 - 1)\hat{i} + (1 - 2)\hat{j} + 3\hat{k} = \hat{i} - \hat{j} + 3\hat{k} \]

### Step 3: Set up the integral
Now we can express the integral for the potential difference:

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

Substituting \( \vec{E} \) and \( d\vec{r} \):

\[ V_A - V_B = -\int_A^B (2\hat{i} + 3\hat{j}) \cdot (\hat{i} - \hat{j} + 3\hat{k}) \, dx \, dy \, dz \]

### Step 4: Calculate the dot product
The dot product \( \vec{E} \cdot d\vec{r} \) is:

\[ \vec{E} \cdot d\vec{r} = (2\hat{i} + 3\hat{j}) \cdot (\hat{i} - \hat{j} + 3\hat{k}) = 2(1) + 3(-1) + 0 = 2 - 3 = -1 \]

### Step 5: Integrate
Since the dot product is a constant, we can integrate over the limits of the path from A to B. The limits for \( x \) are from 1 to 2, and for \( y \) are from 2 to 1. The \( z \) component does not contribute since \( d\vec{r} \) has no \( k \) component.

Thus, we can write:

\[ V_A - V_B = -\int_{1}^{2} -1 \, dx + -\int_{2}^{1} -1 \, dy \]

Calculating these integrals gives:

\[ = -[-1 \cdot (2 - 1)] + -[-1 \cdot (1 - 2)] = -[-1] + -[1] = 1 + 1 = 2 \]

### Step 6: Finalize the potential difference
Thus, the potential difference is:

\[ V_A - V_B = -2 \]

### Conclusion
The potential difference \( V_A - V_B \) is:

\[ \boxed{-1} \]