question_answer

An electric dipole with dipole moment \[\frac{{{p}_{0}}}{\sqrt{2}}\,(\hat{i}+\hat{j})\] is held fixed at the origin O in the presence of an uniform electric field of magnitude \[{{\text{E}}_{0}}.\]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: \[({{\varepsilon }_{0}}\] is permittivity of free space, R >> dipole size)

A) Total electric field at point A is \[{{\vec{E}}_{A}}\,=\,\sqrt{2}{{E}_{0}}\,(\hat{i}+\hat{j})\]

B) Total electric field at point B is \[{{\vec{E}}_{B}}\,=\,0\,\]

C) \[\text{R}\,=\,{{\left( \frac{{{p}_{0}}}{4\pi {{\varepsilon }_{0}}{{E}_{0}}} \right)}^{1/3}}\]

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

Correct Answer: B , C

Solution :

\[{{\text{E}}_{\text{net}}}\] should be \[\bot \]to surface so \[\frac{k{{\text{P}}_{0}}}{{{r}^{3}}}\,={{\text{E}}_{0}}\]

\[\Rightarrow \]   \[r={{\left( \frac{k{{p}_{0}}}{{{\text{E}}_{0}}} \right)}^{\text{1/3}}}\]

\[{{({{\text{E}}_{A}})}_{net}}=\frac{2k{{\text{P}}_{0}}}{{{r}^{3}}}+{{\text{E}}_{0}}=3{{\text{E}}_{0}}\]

\[{{({{\text{E}}_{\text{B}}}\text{)}}_{\text{net}}}=0\]

Solution :

Same as above