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

Options

A. `k ( vec(p) xx vec(r))/(r^(2))`
B. `k(vec(p) xx vec(r))/(x^(2))`
C. `k(vec(p) .(-vec(r)))/(r^(2))`
D. `k(vec(p).vec(r))/(r^(3))`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Dipole Configuration**:
- An electric dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance \(2l\). Let’s denote the dipole moment \(P = Q \cdot 2l\).

2. **Positioning the Dipole**:
- Place the dipole along the axis, with the positive charge at point A and the negative charge at point B. The midpoint O is at a distance \(l\) from both charges.

3. **Identifying the Point of Interest**:
- Let point P be at a distance \(r\) from the midpoint O of the dipole. The angles formed with respect to the dipole axis will be denoted as \(\theta\).

4. **Calculating Distances**:
- The distances from point P to the charges can be expressed as:
- Distance \(BP = r - l \cos \theta\) (from negative charge)
- Distance \(AP = r + l \cos \theta\) (from positive charge)

5. **Electric Potential Due to Each Charge**:
- The electric potential \(V\) at point P due to the positive charge is given by:
\[ V_+ = \frac{1}{4\pi \epsilon_0} \frac{Q}{AP} = \frac{1}{4\pi \epsilon_0} \frac{Q}{r + l \cos \theta} \]
- The electric potential due to the negative charge is:
\[ V_- = \frac{1}{4\pi \epsilon_0} \frac{-Q}{BP} = -\frac{1}{4\pi \epsilon_0} \frac{Q}{r - l \cos \theta} \]

6. **Total Electric Potential at Point P**:
- The total electric potential \(V\) at point P is the sum of the potentials due to both charges:
\[ V = V_+ + V_- = \frac{1}{4\pi \epsilon_0} \left( \frac{Q}{r + l \cos \theta} - \frac{Q}{r - l \cos \theta} \right) \]

7. **Finding a Common Denominator**:
- To combine the fractions, find a common denominator:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot Q \cdot \frac{(r - l \cos \theta) - (r + l \cos \theta)}{(r + l \cos \theta)(r - l \cos \theta)} \]
- This simplifies to:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot Q \cdot \frac{-2l \cos \theta}{r^2 - l^2 \cos^2 \theta} \]

8. **Assuming Small Dipole Approximation**:
- For a small dipole (where \(l\) is much smaller than \(r\)), we can neglect the \(l^2 \cos^2 \theta\) term in the denominator:
\[ V \approx \frac{1}{4\pi \epsilon_0} \cdot \frac{2Q l \cos \theta}{r^2} = \frac{1}{4\pi \epsilon_0} \cdot \frac{P \cos \theta}{r^2} \]

9. **Final Expression**:
- The expression for the electric potential \(V\) at point P due to an electric dipole is:
\[ V = \frac{K \cdot P \cdot \cos \theta}{r^2} \]
- In vector form, this can be expressed as:
\[ V = \frac{K \cdot \mathbf{P} \cdot \mathbf{r}}{r^3} \]
- Where \(K = \frac{1}{4\pi \epsilon_0}\).

### Final Answer:
The value of electric potential at any point due to an electric dipole is given by:
\[ V = \frac{K \cdot \mathbf{P} \cdot \mathbf{r}}{r^3} \]