An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

• Correct answer
  A

  $\mathrm{R}={\left(\frac{{\mathrm{p}}_0}{4\mathrm{\pi }{\mathrm{\epsilon }}_0{\mathrm{E}}_0}\right)}^{\frac{1}{3}}$
• B

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

  Total electric field at point $\mathrm{A}$ is ${\overrightarrow{\mathrm{E}}}_{\mathrm{A}}=\sqrt{2}{\mathrm{E}}_0\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$
• Correct answer
  D

  Total electric field at point $\mathrm{B}$ is ${\overrightarrow{\mathrm{E}}}_{\mathrm{B}}=0$

Solution

Radius of circle, $R\gg$ dipole size
As, circle is equipotential
So, $E_{\mathrm{n}\mathrm{e}\mathrm{t}}$ should be $\perp$ to surface, so $\frac{k{p}_0}{r^3}=E_0\Rightarrow r={\left(\frac{k{p}_0}{E_0}\right)}^{1/3}\Rightarrow \left(A\right)$ is correct
At point $B$ net electric field will be zero.
$E_B=0$
${\left(E_A\right)}_{\mathrm{N}\mathrm{e}\mathrm{t}}=\frac{2k{p}_0}{r^3}+E_0=3E_0$
Electric field at point $A, {\overrightarrow{E}}_A=\frac{3}{\sqrt{2}}{E}_0\left[\widehat{i}+\widehat{j}\right]\Rightarrow \left(C\right)$ is incorrect
${\left(E_B\right)}_{\mathrm{N}\mathrm{e}\mathrm{t}}=0\Rightarrow \left(D\right)$ is correct

Hence, the correct answer is options: 'A', 'D'.