Derivation of the Potential at Any Point on the Axial Line due to an Electric Dipole

The potential at any point on the axial line due to an electric dipole can be derived by considering the superposition principle. We will consider a dipole consisting of two charges, +q and -q, separated by a distance 2a.

Step 1: Defining the Coordinate System and Parameters
- We define a coordinate system with the origin located at the center of the dipole.
- The positive charge +q is located at a distance +a along the x-axis, while the negative charge -q is located at a distance -a along the x-axis.
- We will consider a point P on the axial line at a distance x from the center of the dipole.

Step 2: Deriving the Potential due to +q
- The potential at point P due to the positive charge +q can be calculated using the formula for the potential due to a point charge:
V1 = k*q/(r1)
where k is the electrostatic constant and r1 is the distance between +q and point P.
- In this case, r1 = x - a, as the distance between +q and point P is the difference between their respective x-coordinates.

Step 3: Deriving the Potential due to -q
- Similarly, the potential at point P due to the negative charge -q can be calculated using the same formula:
V2 = k*(-q)/(r2)
where r2 is the distance between -q and point P.
- In this case, r2 = x + a, as the distance between -q and point P is the sum of their respective x-coordinates.

Step 4: Applying the Superposition Principle
- The total potential at point P due to the electric dipole is the sum of the potentials due to +q and -q:
V = V1 + V2
V = k*q/(x - a) + k*(-q)/(x + a)

Step 5: Simplifying the Expression
- To simplify the expression further, we can multiply and divide by a common term, (x^2 - a^2), in the denominators:
V = k*q*(x + a)/(x^2 - a^2) + k*(-q)*(x - a)/(x^2 - a^2)
V = k*q*(x + a - x + a)/(x^2 - a^2)

Step 6: Final Expression
- After simplifying, we obtain the final expression for the potential at any point on the axial line due to an electric dipole:
V = 2k*q*a/(x^2 - a^2)

Conclusion
- The derived expression shows that the potential at any point on the axial line due to an electric dipole depends on the dipole moment (q*a) and the distance between the point and the dipole (x). The potential is inversely proportional to the square of the difference between the square of the distance and the square of the separation between the charges in the dipole.