Question

A solid insulating sphere of radius $R$ is given a charge $Q$. If at a point inside the sphere the potential is $1.5$ times the potential at the surface, this point will be:

• A. at a distance of $2R/3$ form the centre
• B. at the centre
• C. at a distance of $2R/3$ form the surface
• D. data insufficient

Answer 1

The correct option is B at the centre

For Insulating sphere,

$V_{surface}=\frac{RQ}{R}R=\frac{1}{4\pi {ϵ}_0},V=potential$

$V_{inside}=R\cdot \frac{Q}{R}[\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}]$ r=distance from center.

$V_{inside}=1.5V_{surface}$

$R\cdot \frac{Q}{R}[\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}]=1.5\frac{RQ}{R}$

$\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}=1.5$

$\Rightarrow r^2=0$

$\Rightarrow r=0$

Hence, the point is placed at center$(x=0)$.