Derivation of Potential due to a Dipole

The potential due to a dipole can be derived by considering a point P at a distance r from the center of the dipole. The dipole is formed by two equal and opposite charges, q and -q, separated by a distance d.

Introduction
To derive the potential due to a dipole, we will consider the following steps:

1. Determine the potential due to the positive charge at P.
2. Determine the potential due to the negative charge at P.
3. Add the potentials due to both charges to obtain the total potential at P.

1. Potential due to the Positive Charge
The potential due to a point charge at a distance r is given by the equation V+ = k * q / r, where k is the electrostatic constant.

Considering the positive charge q of the dipole at point P, the potential due to this charge is given by V+ = k * q / r+. Here, r+ represents the distance between the positive charge and point P.

2. Potential due to the Negative Charge
Similar to the positive charge, the potential due to a point charge at a distance r is given by the equation V- = k * (-q) / r.

The negative charge -q of the dipole at point P gives rise to a potential given by V- = k * (-q) / r-. Here, r- represents the distance between the negative charge and point P.

3. Total Potential due to the Dipole
To obtain the total potential at point P, we add the potentials due to both charges. Therefore, V = V+ + V-.

Substituting the expressions for V+ and V- into the equation, we get V = k * q / r+ + k * (-q) / r-.

Since the dipole is defined as the separation between the charges, d = r+ + r-, we can express r+ and r- as d/2 and d/2, respectively.

Simplifying the equation further, we have V = k * q / (d/2) - k * q / (d/2), which can be rewritten as V = k * q * (1/d - 1/d) = 0.

Therefore, the total potential at point P due to the dipole is zero.

Conclusion
In conclusion, the derivation of the potential due to a dipole shows that the total potential at a point P, located at a distance r from the center of the dipole, is zero. This result indicates that the electric field due to a dipole is a purely dipolar field, with the positive and negative charges canceling each other's potential contributions.