Derive the expression for the electric potential due to an electric...

Question

Derive the expression for the electric potential due to an electric dipole at an axial point.

Accepted Answer

📌 Derivation:

Setup:

Dipole: charges +q and −q, separated by distance 2a (dipole length).
- Dipole moment: p = q × 2a
- Point P is on the axial line at distance r from centre.

Step 1 — Distance from each charge to P:

r₁ = r − a (from +q)

r₂ = r + a (from −q)

Step 2 — Potential at P:

V = k[+q/r₁ + (−q)/r₂]

V = kq [1/(r−a) − 1/(r+a)]

V = kq [(r+a − r+a) / (r²−a²)]

V = kq [2a / (r²−a²)]

Step 3 — In terms of Dipole Moment (p = 2qa):

V = kp / (r² − a²)

Step 4 — For Short Dipole (r >> a, so a² << r²):

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│ V = kp / r² │

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Note: At equatorial point, V = 0 (positive and negative contributions cancel).