The value of electric potential at any point due to any electric dipole is

Options

A. `k. (vecp xx vecr)/(r^(2))`
B. `k. (vecp xx vecr)/(r^(3))`
C. `k. (vecp .vecr)/(r^(2))`
D. `k. (vecp .vecr)/(r^(3))`

Explanation

To find the value of electric potential at any point due to an electric dipole, we can follow these steps:

### Step 1: Understand the Configuration of the Dipole
An electric dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance of 2L. The dipole moment \( \vec{P} \) is defined as:
\[ \vec{P} = Q \cdot (2L) \]
This dipole moment points from the negative charge to the positive charge.

### Step 2: Identify the Point of Interest
Let’s consider a point P at a distance R from the midpoint of the dipole. The angles made by the line joining the dipole to point P with the dipole axis are denoted by \( \theta \).

### Step 3: Calculate the Distances from Charges to Point P
The distances from the charges to point P can be expressed as:
- Distance from +Q (at point A) to P:
\[ AP = R - L \cos \theta \]
- Distance from -Q (at point B) to P:
\[ BP = R + L \cos \theta \]

### Step 4: Write the Electric Potential Due to Each Charge
The electric potential \( V \) at point P due to a point charge is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]
Thus, the potentials due to the two charges at point P are:
- Due to +Q:
\[ V_1 = \frac{1}{4 \pi \epsilon_0} \frac{Q}{R - L \cos \theta} \]
- Due to -Q:
\[ V_2 = -\frac{1}{4 \pi \epsilon_0} \frac{Q}{R + L \cos \theta} \]

### Step 5: Combine the Potentials
The total electric potential \( V \) at point P is the sum of the potentials due to both charges:
\[ V = V_1 + V_2 = \frac{1}{4 \pi \epsilon_0} \left( \frac{Q}{R - L \cos \theta} - \frac{Q}{R + L \cos \theta} \right) \]

### Step 6: Simplify the Expression
By finding a common denominator:
\[ V = \frac{1}{4 \pi \epsilon_0} \cdot Q \left( \frac{(R + L \cos \theta) - (R - L \cos \theta)}{(R - L \cos \theta)(R + L \cos \theta)} \right) \]
This simplifies to:
\[ V = \frac{1}{4 \pi \epsilon_0} \cdot Q \cdot \frac{2L \cos \theta}{R^2 - (L \cos \theta)^2} \]

### Step 7: Use the Dipole Moment
Substituting \( P = Q \cdot 2L \):
\[ V = \frac{1}{4 \pi \epsilon_0} \cdot \frac{P \cdot \cos \theta}{R^2} \]
For large distances (where \( R \) is much larger than \( L \)), the term \( (L \cos \theta)^2 \) can be neglected:
\[ V \approx \frac{1}{4 \pi \epsilon_0} \cdot \frac{P \cdot \cos \theta}{R^2} \]

### Final Answer
Thus, the electric potential \( V \) at a point due to an electric dipole is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \cdot \frac{P \cdot \cos \theta}{R^2} \]