An electric dipole with dipole moment $\frac{P o}{\sqrt{2}} (\bar{i} + \bar{j})$ is held fixed at the origin ‘o’in the presence of an uniform electric field of magnitude $E_{0}$ . If the potential is constant on a circle of radius R centered at the origins shown in figure ,then the correct statement(s) is/are :

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Total electric field at point A is $\vec{E_{A}} = \sqrt{2} E_{0} (\bar{i} + \bar{j})$

Total electric field at point B is $E_{B} = 0$

$R = {(\frac{P o}{4 π ε_{0} E_{0}})}^{1 / 3}$

The magnitude of total electric field on any two points of the circle will be same

answer is B, C.

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Detailed Solution

At $A + B, \vec{E}$ due to dipole tangential also at B, if the other field also had a component in radial direction, then this component would contribute to the net tangential component at A (since the other field is given to be uniform), which is not allowed since the given sphere is equipotential.

$\Rightarrow E_{o t h e r} = \frac{1}{4 π ε_{0}} . \frac{P_{0}}{R^{3}} (\frac{\bar{i} + \bar{j}}{\sqrt{2}})$