A small electric dipole is kept on the axis of a uniformly charged ring at distance $R / \sqrt{2}$ from the centre of the ring. The direction of the dipole moment is along the axis. The dipole moment is P. charge of the ring is Q and radius of the ring is R. The force on the dipole is nearly ____.

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Detailed Solution

Force acting on a dipole placed in a non-uniform external field will be given by $F = P \frac{dE}{dx}$

$E (x) = \frac{kQx}{{(x^{2} + R^{2})}^{3 / 2}}$ ,where x is the separation between the center of the ring to the observation point.

$\frac{dE}{dx} = kQ [\frac{(x^{2} + R^{2}) - \frac{3}{2} {(x^{2} + R^{2})}^{1 / 2} 2 x . x}{{(x^{2} + R^{2})}^{3}}]$

$= kQ \sqrt{(x^{2} + R^{2})} [\frac{x^{2} + R^{2} - 3 x^{2}}{{(x^{2} + R^{2})}^{3}}]$

$= kQ \sqrt{(x^{2} + R^{2})} [\frac{R^{2} - 2 x^{2}}{{(x^{2} + R^{2})}^{3}}]$

at $x = \frac{R}{\sqrt{2}}$ , $\frac{dE}{dx} = 0$

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