Answer to Question #212980 in Electricity and Magnetism for Anita

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

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

$r=b<a$

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

Where

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

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

b>a

$E_{o}.A=\frac{q}{\epsilon_0},E_{o}=\frac{q}{4\pi \epsilon _0b^3}$

Potential

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

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

$V_a=-[\smallint _{inf}^bE_o.dr+\int_b^aE_{in}.dr]$

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

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

b=a

$V_{in}=\frac{3Q}{8\pi \epsilon_0a}$

$∆V=V_{in}{(r=a)}-V_{in}{(r=0)}$

$∆V=\frac{-Q}{8\pi\epsilon_0 a}$