12.4 Potential due to a Ring

  Electrostatics

(i) Potential at the centre of uniformaly charged ring:

Potential due to a small element $dq$ $$dV = \frac{{Kdq}}{R}$$

$\therefore $ Net potential: $V = \int {\frac{{K\;dq}}{R}} $

$\therefore \;V = \frac{K}{R}\;\int {dq = \frac{{Kq}}{R}} $

(ii) For non uniformaly charged ring potential at the center is: $$V = \frac{{K{q_{total}}}}{R}$$

(iii) Potential due to half ring at center is: $$V = \frac{{Kq}}{R}$$

(iv) Potential at the axis of a ring:

Calculation of potential at a point on the axis which is a distance $X$ from center of uniformly charged $($ total charge $Q)$ ring of radius $R.$

consider an element of charge $dq$ on the ring.

Potential at point $P$ due to charge $dq$ will be $$dV = \frac{{K\left( {dq} \right)}}{{\sqrt {{R^2} + {X^2}} }}$$

$\therefore $ Net potential at point $P$ due to all such element will be: $$V = \int {dV} = \frac{{KQ}}{{\sqrt {{R^2} + {X^2}} }}$$