Question

A point negative charge $-Q$ is placed at a distance $r$ from a dipole with dipole moment $P$ in the $\text{x-y}$ plane as shown in figure

The $\text{x-}$component of force acting on the charge $-Q$ is

• A. $-\frac{2PkQ}{r^3}\cos \theta \hat{i}$
• B. $\frac{2PkQ}{r^3}\cos \theta \hat{i}$
• C. $\frac{PkQ}{r}\cos \theta \hat{i}$
• D. $-\frac{PkQ}{r}\cos \theta \hat{i}$

Answer 1

The correct option is A

$-\frac{2PkQ}{r^3}\cos \theta \hat{i}$

Resolve components of dipole moment $\text{P}$ along the $\text{x-}$axis as shown in the figure.


Electric field due to the component $P\cos \theta$ at distance $r$,

${\overrightarrow{E}}_1=\frac{2kP\cos \theta }{r^3}\hat{i}$

Electric field due to the component $P\sin \theta$,

${\overrightarrow{E}}_2=\frac{kP\sin \theta }{r^3}(-\hat{j})$

Since charge $-Q$ is on the axial line at distance $r$.

So, the force, $F_x$, along the $\text{x-}$axis on charge $-Q$ will be only due to $P\cos \theta$ component of dipole moment which is

${\overrightarrow{F}}_x=-Q{\overrightarrow{E}}_1$

Substituting the value of ${\overrightarrow{E}}_1$,

$\Rightarrow {\overrightarrow{F}}_x=\frac{-2PkQ}{r^3}\cos \theta \hat{i}$

Hence, option (c) is the correct option.