Question

A charge Q is uniformly distributed throughout a sphere of radius a. Taking the potential 

at infinity as zero, the potential at r =b < a is

Answer 1

Gauss law

$\smallint E.A=\frac{Q}{\epsilon_0}$

r=b<a

$E.(4\pi b^2)=\frac{\rho\frac{4\pi b^2}{3}}{\epsilon_0}$

$\rho=\frac{Q}{\frac{4\pi a^3}{3}}$

$E_{in}=\frac{qb}{4\pi\epsilon_0a^3}$

b>a

$E.A=\frac{q}{\epsilon_0}$

$E_0=\frac{q}{4\pi\epsilon_0b^2}$

Potential

$V_a-V_{infinite}=\smallint_{inf}^aE.dr$

$V_a=-\smallint_{inf}^aE.dr$

$V_a=-[\smallint_{inf}^bE_0.dr+\smallint_{a}^bE_{in}.dr]$

$V_a=-[\smallint_{inf}^b\frac{kQ}{r^2}dr+\smallint_{b}^a \frac{kQr}{R^3}dr]$

$V_{in}=\frac{kQ(3a^2-b^2)}{2a^3}$

$b=a;$

$V_{in}=\frac{3Q}{8\pi\epsilon_0 a}$

$∆V=V_{in}(r=a)-V_{in}(r=0)$

$∆V=\frac{-Q}{8\pi\epsilon_0 a}$