A dipole of dipole moment `vec(P)=2hat(i)-3hat(j)+4hat(k)` is placed at point A (2, -3, 1). The electric potential due to this dipole at the point B (4, -1, 0) is equal to (All the parameters specified here are in S.L. units) ____`xx 10^(9)` volts

Answer

To find the electric potential \( V \) at point B due to the dipole at point A, we can follow these steps:

### Step 1: Identify the dipole moment and the coordinates of points A and B
The dipole moment is given as:
\[ \vec{P} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]
Point A has coordinates \( (2, -3, 1) \) and point B has coordinates \( (4, -1, 0) \).

### Step 2: Calculate the position vector \( \vec{r} \) from point A to point B
The position vector \( \vec{r} \) from point A to point B is given by:
\[ \vec{r} = \vec{B} - \vec{A} = (4 - 2)\hat{i} + (-1 + 3)\hat{j} + (0 - 1)\hat{k} = 2\hat{i} + 2\hat{j} - 1\hat{k} \]

### Step 3: Calculate the magnitude of the position vector \( |\vec{r}| \)
The magnitude of \( \vec{r} \) is calculated as follows:
\[ |\vec{r}| = \sqrt{(2)^2 + (2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 4: Calculate \( |\vec{r}|^3 \)
Now, we calculate \( |\vec{r}|^3 \):
\[ |\vec{r}|^3 = 3^3 = 27 \]

### Step 5: Calculate the dot product \( \vec{P} \cdot \vec{r} \)
The dot product \( \vec{P} \cdot \vec{r} \) is computed as follows:
\[ \vec{P} \cdot \vec{r} = (2\hat{i} - 3\hat{j} + 4\hat{k}) \cdot (2\hat{i} + 2\hat{j} - 1\hat{k}) \]
Calculating the dot product:
\[ = 2 \cdot 2 + (-3) \cdot 2 + 4 \cdot (-1) = 4 - 6 - 4 = -6 \]

### Step 6: Use the formula for electric potential \( V \)
The formula for the electric potential \( V \) due to a dipole is given by:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot \frac{\vec{P} \cdot \vec{r}}{|\vec{r}|^3} \]
Given that \( \frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \):
\[ V = 9 \times 10^9 \cdot \frac{-6}{27} \]

### Step 7: Simplify the expression for \( V \)
Calculating the above expression:
\[ V = 9 \times 10^9 \cdot \left(-\frac{2}{9}\right) = -2 \times 10^9 \, \text{volts} \]

### Final Answer
The electric potential at point B due to the dipole is:
\[ \boxed{-2 \times 10^9 \, \text{volts}} \]