Question

A conducting sphere of radius R, carrying charge Q, lies inside an uncharged conducting shell of radius 2R. If they are joined by a metal wire, then

• (A) Q/3 amount of charge will flow from the sphere to the shell
• (B) 2Q/3 amount of charge will flow from the sphere to the shell
• (C) Q amount of charge will flow from the sphere to the shell
• (D) K Q²/4R amount of heat will be produced

Answer 1

The Correct Option is D

Explanation:
Capacitance of the capacitor (shell) having radius $2\mathrm{R},C_2=4\pi {\epsilon }_0(2R)$Potential at surface of sphere, $V_1=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{R}$Potential at the surface of the shell, $V_2=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{2R}$Hence, sphere is at high potential and shell is at low potential. So, charge will flow from sphere to shell.Energy stored in ${\mathrm{C}}_1,U_1=\frac{Q^2}{2C_1}$Energy stored in ${\mathrm{C}}_2,U_2=\frac{Q^2}{2C_2}$Heat liberated is$H=U_1-U_2=\frac{Q^2}{2}[\frac{1}{4\pi {\epsilon }_{0}R}-\frac{1}{4\pi {\epsilon }_0\times 2R}]$$H=\frac{1}{4\pi {\epsilon }_0}\frac{Q^2}{2R}[1-\frac{1}{2}]=\frac{k{Q}^2}{4R}$Here, $k=\frac{1}{4\pi {\epsilon }_0}$

Answer 2

Energy Stored in a Charged Capacitor

The process of charging a capacitor is equivalent to that of transferring charge from one plate to the other plate of the capacitor. Some work must be done to charge a capacitor. This work is stored as electrostatic potential energy in the capacitor.

The formula for energy stored in a capacitor is given by

U = Q²/2C  or U = CV²/2 or U = QV/2

Where

• U is the electrostatic potential energy of the capacitor
• Q is the amount of charge on the plates of the capacitor
• V is the potential difference across the plates
• C is the capacitance

Electrostatic Energy Density

Energy stored per unit volume of a capacitor is known as electrostatic energy density.

The formula for the energy density of a capacitor is given by

Energy density, U_d = ϵ₀E² / 2

Its SI unit is Joule per cubic meter (J m^-3).