Q.Define electric potential at a point. Obtain an expression for the electrostatic potential due to an electric dipole. OR Define capacitance of a capacitor. Obtain an expression for the capacitance of a parallel plate capacitor and hence explain the effect of a dielectric on its capacitance.

Electric potential is the work done per unit charge to bring a test charge from infinity to a point; superposing the potentials of a dipole's two point charges and using the far-field (dipole) approximation gives $V=\dfrac{p\cos\theta}{4\pi\varepsilon_0 r^2}$.

Definition: The electric potential at a point in an electric field is defined as the amount of work done in bringing a unit positive test charge from infinity to that point, without any acceleration (quasi-statically):

$V = \frac{W}{q_0} = -\int_\infty^r \vec E\cdot d\vec r$

It is a scalar quantity, measured in volts.

Potential due to an electric dipole:

Consider a dipole with charges $-q$ at A and $+q$ at B, separated by $2a$, dipole moment $p=q(2a)$. Let $P$ be a point at distance $r$ from the centre $O$ of the dipole, with the line $OP$ making angle $\theta$ with the dipole axis (from $-q$ to $+q$).

Let $r_1$ = distance from $+q$ to $P$, $r_2$ = distance from $-q$ to $P$.

By superposition, the potential at P is the sum of potentials due to each charge:

$V = \frac{1}{4\pi\varepsilon_0}\left(\frac{q}{r_1}-\frac{q}{r_2}\right) = \frac{q}{4\pi\varepsilon_0}\left(\frac{r_2-r_1}{r_1 r_2}\right)$

For a point far from the dipole ($r \gg a$), using the geometry (dropping perpendiculars from A and B onto OP):

$r_2 - r_1 \approx 2a\cos\theta, \qquad r_1 r_2 \approx r^2$

Substituting: …