Question

Obtain an expression for the electric potential due to a small dipole of dipole moment \( \vec{p} \), at a point \( \vec{r} \) from its centre, for much larger distances compared to the size of the dipole.

Answer 1

Expression for Electric Potential:} The electric potential due to a dipole is the sum of the potentials due to the charges \( +q \) and \( -q \) separated by distance \( 2a \): \[ V = \frac{1}{4\pi\epsilon_0} \left( \frac{q}{r_1} - \frac{q}{r_2} \right) \] Using the geometry of the dipole, we have: \[ r_1^2 = r^2 + a^2 - 2ar \cos \theta, \quad r_2^2 = r^2 + a^2 + 2ar \cos \theta \] For \( r \gg a \), using binomial expansion and retaining first-order terms: \[ \frac{1}{r_1} \approx \frac{1}{r} \left( 1 - \frac{a \cos \theta}{r} \right) \] \[ \frac{1}{r_2} \approx \frac{1}{r} \left( 1 + \frac{a \cos \theta}{r} \right) \] Substituting these into the potential equation: \[ V = \frac{q}{4\pi\epsilon_0} \cdot \frac{2a \cos \theta}{r^2} \] Since the dipole moment \( p = 2qa \), we get: \[ V = \frac{p \cos \theta}{4\pi\epsilon_0 r^2} \]