A solid non-conducting sphere of…

A solid non-conducting sphere of radius R carries a volume charge density ρ=ρ0 r/R for r≤ R, where r is the distance from the centre (ρ0>0 is a…

One or more options correct

A solid non-conducting sphere of radius

R

carries a volume charge density

\rho=\rho_0\,\dfrac{r}{R}

for

r\le R

, where

r

is the distance from the centre (

\rho_0>0

is a constant), and no charge outside. Take the potential to be zero at infinity. Which of the following statements is/are correct?

A. The electric field at $r=R/2$ is one-fourth of the field at the surface.
B. The potential at the centre is $\dfrac{4}{3}$ times the potential at the surface.
C. The electrostatic field energy stored outside the sphere is $7$ times that stored inside it.
D. The field at $r=R/2$ equals the field at that point of a uniformly charged sphere of the same radius and the same total charge.

Show answer and solution

Answer: A, B, C

Charge within radius

r

q(r)=\int_0^r\rho_0\frac{r'}{R}4\pi r'^2dr'=\dfrac{\pi\rho_0r^4}{R}

, so

Q=\pi\rho_0R^3

and by Gauss's law

E(r)=\dfrac{\rho_0r^2}{4\varepsilon_0R}

inside. (A)

E\propto r^2

, so

E(R/2)=E(R)/4

: correct. (B)

V(R)=\dfrac{Q}{4\pi\varepsilon_0R}=\dfrac{\rho_0R^2}{4\varepsilon_0}

and

V(0)=V(R)+\int_0^RE\,dr=\dfrac{\rho_0R^2}{4\varepsilon_0}+\dfrac{\rho_0R^2}{12\varepsilon_0}=\dfrac{\rho_0R^2}{3\varepsilon_0}

, ratio

4/3

: correct. (C) Energy density

\tfrac12\varepsilon_0E^2

: inside

U_i=\dfrac{\pi\rho_0^2R^5}{56\varepsilon_0}

, outside

U_o=\dfrac{\pi\rho_0^2R^5}{8\varepsilon_0}

, ratio

7

: correct. (D) A uniform sphere gives

\dfrac{Qr}{4\pi\varepsilon_0R^3}=\dfrac{\rho_0R}{8\varepsilon_0}

R/2

, whereas here

E(R/2)=\dfrac{\rho_0R}{16\varepsilon_0}

, half as large: incorrect.

Original question written in the JEE Advanced 2023 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake