Question

An electric dipole with dipole moment $\frac{p_0}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)$ is held fixed at the origin O in the presence of an uniform electric field of magnitude E₀. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are:
(${\epsilon }_0$ is permittivity of free space, R >> dipole size)

• A. $R={\left(\frac{P_0}{4\pi {\epsilon }_{0E_0}}\right)}^{{}^{1/3}}$
• B. The magnitude of total electric field on any two points of the circle will be same
• C. Total electric field at point A is ${\vec{E}}_A=\sqrt{2}{E}_0\left(\hat{\hat{i}}+\hat{j}\right)$
• D. Total electric field at point B is ${\vec{E}}_B=0$

Answer 1

Answer: (A), (D)

$\left(1\right)\overrightarrow{P}=\frac{P_0}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)$

E.F. at B along tangent should be zero since circle is equipotential.

So, $E_0=\frac{K|\vec{P}|}{R^3}\&E_B=0$

So, $R^3=\frac{K{P}_0}{E_0}=\left(\frac{P_0}{4\pi {\in }_0{E}_0}\right)$

So $R={\left(\frac{P_0}{4\pi {\in }_0{E}_0}\right)}^{1/3}$

So, (1) is correct
(2) Because $E_0$ is uniform & due to dipole E.F. is different at different points, so magnitude of total E.F. will also be different at different points.
So, (2) is incorrect

$\left(3\right)E_A=\frac{2KP}{R^3}+\frac{KP}{R^3}=3\frac{KP}{R^3}\frac{P_0}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)$

So,t(3) is wrong

$\left(4\right)E_B=0$

so, (4) is correct