Name: Continuous Charge Distribution
Uploaded: 2019-07-19
Duration: 5 min 39 s
Description: This video will help us understand the necessity of continuous charge distribution and learn about the concept of linear charge density. Learn more.

Question

Two concentric hollow conducting spheres of radius $a$ and $b (b > a)$ contains charges $Q_{a}$ and $Q_{b}$ , respectively. If they are connected by a conducting wire then find out

$(i)$ Final charges on inner and outer spheres.

$(ii)$ The heat produced during the process.

Accepted Answer

$(i)$ Whenever any shell is put inside the other shell of a bigger radius and connected through the wire, the charge flows to the surface of the outer shell, this may be proved by the discussion given below. Let charge $q$ flows through the wire.

Potentials will be equal, $V_{A} = V_{B}$

$\frac{k (Q_{a} - q)}{a} + \frac{k (Q_{b} + q)}{b} = \frac{k (Q_{b} + q)}{b} + \frac{k (Q_{a} - q)}{b}$

$\Rightarrow q = Q_{a}$ .

Therefore, the final charge on the inner surface is zero and on the outer surface is $Q_{a} + Q_{b}$ .

$(ii)$ The capacitance of shells $C_{a} = 4 π ε_{0} a$ and $C_{b} = 4 π ε_{0} b$ .

Therefore, loss of electric potential energy will be equal to the heat produced

$U_{i} = \frac{{Q_{a}}^{2}}{2 C_{a}} + \frac{{Q_{b}}^{2}}{2 C_{b}} = \frac{{Q_{a}}^{2}}{8 π ε_{0} a} + \frac{{Q_{b}}^{2}}{8 π ε_{0} b}$

$U_{f} = 0 + \frac{{(Q_{a} + Q_{b})}^{2}}{8 π ε_{0} b}$

Heat lost $= \frac{{Q_{a}}^{2}}{8 π ε_{0} a} + \frac{{Q_{b}}^{2}}{8 π ε_{0} b} - \frac{{(Q_{a} + Q_{b})}^{2}}{8 π ε_{0} b}$

$\Rightarrow \frac{{Q_{a}}^{2}}{8 π ε_{0}} (\frac{b - a}{a b}) + \frac{Q_{a} Q_{b}}{4 π ε_{0} b}$ .

Two concentric hollow conducting spheres of radius $a$ and $b (b > a)$ contains charges $Q_{a}$ and $Q_{b}$ , respectively. If they are connected by a conducting wire then find out

$(i)$ Final charges on inner and outer spheres.

$(ii)$ The heat produced during the process.

Important Questions on Electrostatics

Learn and Prepare for any exam you want !

Two concentric hollow conducting spheres of radius a and bb>a contains charges Qa and Qb, respectively. If they are connected by a conducting wire then find out i Final charges on inner and outer spheres. ii The heat produced during the process.

Important Questions on Electrostatics

Learn and Prepare for any exam you want !

Two concentric hollow conducting spheres of radius $a$ and $b (b > a)$ contains charges $Q_{a}$ and $Q_{b}$ , respectively. If they are connected by a conducting wire then find out

$(i)$ Final charges on inner and outer spheres.

$(ii)$ The heat produced during the process.