Question

A $4\, \mu \,F$ capacitor is charged by a $200 \,V$ battery. It is then disconnected from the supply and is connected to another uncharged $2\, \mu \,F$ capacitor. During the process, loss of energy $( in \,J )$ is

• A. $3.43 \times 10^{-2}$
• B. $2.67 \times 10^{-2}$
• C. $2.67 \times 10^{-4}$
• D. $3.43 \times 10^{-4}$

Answer 1

Answer: (B)

Charge stored at the capacitor
$q=C_{1} V_{1}=4 \times 200=800\, \mu\, C$
When this capacitor is connected with a uncharged capacitor, then common potential on both capacitors
$V=\frac{C_{1} \,V_{1}+C_{2}\, V_{2}}{C_{1}+C_{2}}$
$=\frac{800+0}{4+2}=\frac{800}{6} V$
Loss in energy $=$ Initial energy $-$ Final energy
$=\frac{1}{2} C_{1} V_{1}^{2}-\frac{1}{2}\left(C_{1}+C_{2}\right) V^{2} $
$=\frac{1}{2} \times 4 \times 10^{-6} \times(200)^{2}-\frac{1}{2}(4+2) \times 10^{-6} \times\left(\frac{800}{6}\right)^{2} $
$=2 \times 10^{-6} \times 4 \times 10^{4}-\frac{3 \times 10^{-6} \times 64 \times 10^{4}}{36} $
$=8 \times 10^{-2}-\frac{64}{12} \times 10^{-2} $
$=8 \times 10^{-2}-5.33 \times 10^{-2} $
$=267 \times 10^{-2} \,J$