A network of four \( 9\,\mu\mathrm{F} \) capacitors is connected to a 500 V supply, as shown. In the context of this, identify the correct statements.
(A) \( C_4 \) is connected in parallel to the combination of \( C_1, C_2 \) and \( C_3 \).
(B) The equivalent capacitance of the combination \( C_1, C_2, C_3 \) and \( C_4 \) is \( 12\,\mu\mathrm{F} \).
(C) The charge on \( C_4 \) is \( 4.5 \times 10^{-3}\,\mathrm{C} \).
(D) The charge stored on \( C_1 \) and \( C_4 \) is the same.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
From the figure, \( C_1 \) (A to B), \( C_2 \) (B to C) and \( C_3 \) (C to D) are in series between A and D, and \( C_4 \) is connected directly between A and D, across the 500 V supply.
\( \displaystyle C_{123} = \frac{9}{3} = 3\,\mu\mathrm{F} \)
\( \displaystyle C_{\mathrm{eq}} = C_{123} + C_4 = 3 + 9 = 12\,\mu\mathrm{F} \)
\( \displaystyle Q_4 = C_4 V = 9\times10^{-6} \times 500 = 4.5\times10^{-3}\,\mathrm{C} \)
\( \displaystyle Q_1 = Q_2 = Q_3 = C_{123} V = 3\times10^{-6} \times 500 = 1.5\times10^{-3}\,\mathrm{C} \)
(A) \( C_4 \) is connected in parallel to the combination of \( C_1, C_2 \) and \( C_3 \). — Correct. \( C_4 \) and the series chain both connect A to D, so they are in parallel.
(B) The equivalent capacitance of the combination \( C_1, C_2, C_3 \) and \( C_4 \) is \( 12\,\mu\mathrm{F} \). — Correct. \( 3\,\mu\mathrm{F} + 9\,\mu\mathrm{F} = 12\,\mu\mathrm{F} \).
(C) The charge on \( C_4 \) is \( 4.5 \times 10^{-3}\,\mathrm{C} \). — Correct. The full 500 V is across \( C_4 \), giving \( 4.5\times10^{-3}\,\mathrm{C} \).
(D) The charge stored on \( C_1 \) and \( C_4 \) is the same. — Incorrect. \( C_1 \) carries the series charge \( 1.5\times10^{-3}\,\mathrm{C} \), only one third of the charge on \( C_4 \).