Question

Three uncharged capacitors of capacities $C_1 , C_2$ and $C_3$ are connected as shown in the figure. A, B and C are at potentials $V_1,\, V_2$ and $V_3$ respectively. The potential at ‘O' is

• A. $\frac{C_{1} V_{1}+C_{2} V_{2}+C_{3} V_{3}}{C_{1}+C_{2}+C_{3}}$
• B. $\frac{C_{1} V_{1}+C_{2} V_{2}-C_{3} V_{3}}{C_{1}+C_{2}+C_{3}}$
• C. $\frac{C_{1} V_{1}-C_{2} V_{2}-C_{3} V_{3}}{C_{1}+C_{2}+C_{3}}$
• D. $\frac{C_{1} V_{1}-C_{2} V_{2}+C_{3} V_{3}}{C_{1}+C_{2}+C_{3}}$
• E. Zero

Answer 1

Answer: (A)

Total charge in the capacitors (accumulated at point $O$ )
$Q_{\text {total }}=C_{1} V_{1}+C_{2} V_{2}+C_{3} V_{3}$
Total capacitance of the combination,
$C_{\text {total }}=C_{1}+C_{2}+C_{3}$ (w.r.t. point $\left. O\right)$
So, potential at $O$,
$V = \frac{Q_{\text{total}}}{C_{\text{total}}}$
$=\frac{C_{1} V_{1}+C_{2} V_{2}+C_{3} V_{3}}{C_{1}+C_{2}+C_{3}}$