Question

Three charges are kept on a straight line as shown in figure. With $A$ and $B$ denoting arbitrary constants, the potential at a far-off point $(r>>a)$ on the axis is given by

• A. $\frac{A}{r}$
• B. $\frac{A}{r^2}$
• C. $\frac{A}{r^3}$
• D. $\frac{A}{r^2}+\frac{B}{r^3}+.....$

Answer 1

The correct option is B $\frac{A}{r^2}$

From the given diagram,

Total charge $Q=-2q+q+q=0\text{C}$

Since, the total charge is zero, the value of dipole moment $\overrightarrow{p}$ is independent of origin.


Dipole moment, $\overrightarrow{p}=\sum _{i=1}^3{q}_i{r}_i$

$\Rightarrow \overrightarrow{p}=q\times 0+(-2q)\times a+q\times 3a$

$\Rightarrow \overrightarrow{p}=-2qa+3qa=qa$

$\therefore \overrightarrow{p}\ne 0$

Potential of electric dipole , $V=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{r^2}$

Substituting the value of $p$ into the above equation we get,

$V=\frac{1}{4\pi {\epsilon }_0}\frac{qa\cos \theta }{r^2}$

along the axis $(\theta =0^{\circ })\therefore \cos \theta =1$

$\Rightarrow V\propto \frac{A}{r^2}$

Hence, option (b) is the correct answer.