Electric potential due to a dipole at a position `vec r` from its centre is: where `(K = 1/(4pi epsilon_0)`

Options

A. `(Kvec(p).vec(r))/(r^(3))`
B. `(K.vec(p).vec(r))/(r^(2))`
C. `(K.vec(P)xxvec(r))/(r^(3))`
D. `(K.vec(p)xxvec(r))/(r^(2))`

Answer

To find the electric potential \( V \) due to a dipole at a position \( \vec{r} \) from its center, we can follow these steps:

### Step 1: Understand the Dipole Configuration
A dipole consists of two equal and opposite charges, \( +q \) and \( -q \), separated by a distance \( 2a \). The dipole moment \( \vec{p} \) is defined as:
\[ \vec{p} = q \cdot 2a \]

### Step 2: Define the Position
Let’s denote the position from the center of the dipole to the point where we want to calculate the potential as \( r \). The distance from the positive charge to the point is \( r - a \) and from the negative charge to the point is \( r + a \).

### Step 3: Calculate the Potential Due to Each Charge
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = K \frac{q}{r} \]
where \( K = \frac{1}{4\pi \epsilon_0} \).

1. **Potential due to the positive charge \( +q \)** at distance \( r - a \):
\[ V_1 = K \frac{q}{r - a} \]

2. **Potential due to the negative charge \( -q \)** at distance \( r + a \):
\[ V_2 = -K \frac{q}{r + a} \]

### Step 4: Total Potential at Point \( A \)
The total electric potential \( V \) at point \( A \) is the sum of the potentials due to both charges:
\[ V = V_1 + V_2 = K \frac{q}{r - a} - K \frac{q}{r + a} \]

### Step 5: Simplify the Expression
Factoring out \( Kq \):
\[ V = Kq \left( \frac{1}{r - a} - \frac{1}{r + a} \right) \]
To combine the fractions, find a common denominator:
\[ V = Kq \left( \frac{(r + a) - (r - a)}{(r - a)(r + a)} \right) = Kq \left( \frac{2a}{r^2 - a^2} \right) \]

### Step 6: Relate to Dipole Moment
Since \( p = q \cdot 2a \), we can express \( 2a \) as \( \frac{p}{q} \):
\[ V = K \frac{p}{r^2} \quad \text{(for } r \gg a\text{)} \]

### Final Result
Thus, the electric potential \( V \) due to a dipole at a distance \( r \) from its center is given by:
\[ V = -K \frac{p \cdot \cos \theta}{r^2} \]
where \( \theta \) is the angle between the dipole moment and the line joining the dipole to the point where the potential is being calculated.