Question

Question 2.36: A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.

Class 12 - Physics - Electrostatic Potential And Capacitance Electrostatic Potential And Capacitance Page-92

Answer 1

potential on inner small sphere is
potential at A = potential due to own self + potential due to sphere B
e.g., $V_{A}=V_{AA}+V_{AB}$
$V_{AA}=\frac{1}{4\pi\epsilon_0}\frac{q_A}{r_A}\\\\=\frac{1}{4\pi\epsilon_0}\frac{q_1}{r_1}$

$V_B=\frac{1}{4\pi\epsilon_0}\frac{q_B}{r_B}\\\\=\frac{1}{4\pi\epsilon_0}\frac{q_2}{r_2}$

now, $V_A=\frac{1}{4\pi\epsilon_0}\left[\begin{array}{c}\frac{q_1}{r_1}+\frac{q_2}{r_2}\end{array}\right]$

similarly, the potential on outer shell B is
$V_B=V_{BB}+V_{BA}\\\\V_B=\frac{1}{4\pi\epsilon_0}\left[\begin{array}{c}\frac{q_2}{r_2}+\frac{q_1}{r_2}\end{array}\right]$

$so,V_A-V_B=\frac{q_1}{4\pi\epsilon_0}\left[\begin{array}{c}\frac{r_2-r_1}{r_1r_2}\end{array}\right]$

here, $r_2>r_1,so\:V_A>V_B$ e.g., inner sphere A at higher potential than outer conducting shell B, do any value of charge $q_$.so,when inner sphere A is connected to outer shell B, then charge will fpow from inner sphere A to outer spherical shell B, until both attain same electrical potential.
e.g., $V_A-V_B=0\:or\:q_1=0$ where $r_2>r_1$
so, charge $q_1$ given to sphere A will flow on the spherical shell B, no matter what the charge on the shell B.