Q.Define electric potential. Obtain an expression of electric potential due to an electric dipole at any point (r, theta). Draw necessary diagram. OR What is capacitor? Derive an expression of capacitance of the spherical capacitor. Draw necessary diagram.

Electric potential is the work done per unit charge to bring a test charge from infinity to a point; for a dipole this potential falls off as $1/r^2$ and depends on the angle from the dipole axis.

Definition of electric potential:

The electric potential $V$ 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 (i.e. quasi-statically, against the electric force):

$V=\dfrac{W}{q_0}$

Electric potential due to a dipole at point (r, θ):

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

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

The potential at P due to the two point charges (superposition):

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

For a point far from the dipole ($r\gg a$, the "short dipole" approximation), using the geometry (dropping perpendiculars from ±q onto the line OP):

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

So: $r_2-r_1\approx 2a\cos\theta$, and $r_1 r_2\approx r^2$.

Substituting:

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