Question

Four short dipoles each of dipole moment p are placed at the vertices of a square of side a. The direction of the dipole moments are shown in the figure.

• A. Electric field at O is $\frac{\sqrt{2}p}{2\pi {ϵ}_0{a}^3}$
• B. Electric field at O is $\frac{\sqrt{2}p}{\pi {ϵ}_0{a}^3}$
• C. Electric potential at O is zero.
• D. Net Electric dipole moment is 2p.

Answer 1

Answer: (B), (D)

The correct options are
B

Electric field at O is $\frac{\sqrt{2}p}{\pi {ϵ}_0{a}^3}$

D

Net Electric dipole moment is 2p.

$E_A\to$ Electric field at axis
$E_E\to$ Electric field at equator
$E_{A2}\&E_{A4}$ will cancel out each other
$E_{A1}\&E_{A3}$ will cancel each other.

$E_{net}=\sqrt{(E_{E2}+E_{E4}{)}^4+(E_{E1}+E_{E3}{)}^2}=\sqrt{2{\left[\frac{\frac{2kp}{\sqrt{2}}}{(\frac{a}{\sqrt{2}}{)}^3}\right]}^2}=\frac{\sqrt{2}p}{\pi {ϵ}_0{a}^3}{P}_{net}={\overrightarrow{p}}_1+{\overrightarrow{p}}_2+{\overrightarrow{p}}_3+{\overrightarrow{p}}_4=p\hat{i}+p\hat{i}+p\hat{j}-p\hat{j}=2p\hat{i}$