Question

The electrostatic potential $\left({\varphi }_r\right)$ of a spherical symmetric system kept at origin is shown in the adjacent figure, and given as


${\varphi }_r=\frac{q}{4\pi {ϵ}_{0}r}...(r⩾R_0)$,
${\varphi }_r=\frac{q}{4\pi {ϵ}_0{R}_0}...(r\le R_0)$.
Which of the following options is/are correct?

• A. For spherical region $r<R_0,$ the total electrostatic energy stored is zero
• B. Within $r=2R_0,$ the total charge is $q$.
• C. There will be no charge anywhere except at $r=R_0.$
• D. The electric field is discontinuous at $r=R_0.$

Answer 1

Answer: (A), (B), (C), (D)

The electric field is discontinuous at $r=R_0.$

The nature of the graph showing the variation of the potential ${\varphi }_r$ with distance r corresponds to that due to a uniformly charged spherical conducting shell of radius $R_0$.
Inside the shell $(r<R_0$, electric field $E=0$, so the total electrostatic energy stored is zero.
Total charge $q$ will be on the surface of the shell $(r=R_0)$ and nowhere else.
Within the concentric spherical shell of radius $r=2R_0,$ total charge contained is $q$.
Electric field with the shell is zero and at the surface $(r=R_0)$, it suddenly changes to $\frac{1}{4\pi {ϵ}_0}\frac{q}{R_0^2}$, so there is discountinuity at $r=R_0$.
Hence, the correct options are (a), (b), (c) and (d).