A network of four 9 μF 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 μF.
(C) The charge on $C_4$ is $4.5×10^{-3} $C.
(D) The charge stored on $C_1$ and $C_4$ is the same.

Choose the correct answer from the options given below:

(A), (B) and (C) only

The correct answer is Option (2) → (A), (B) and (C) only

Given:

Four capacitors each $C = 9 \, \mu F$ connected to a $500 \, V$ source.

Analysis:

1. $C_{1}, C_{2}, C_{3}$ form a series combination between nodes A and D.
Their equivalent capacitance is:

$\frac{1}{C_{eq}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}}$

$\frac{1}{C_{eq}} = \frac{1}{9} + \frac{1}{9} + \frac{1}{9}$

$\frac{1}{C_{eq}} = \frac{3}{9} = \frac{1}{3}$

$C_{eq} = 3 \, \mu F$

2. $C_{4}$ is connected directly across A and D, i.e. in parallel with the series combination of $C_{1}, C_{2}, C_{3}$.

Total equivalent capacitance:

$C_{total} = C_{4} + C_{eq} = 9 + 3 = 12 \, \mu F$

3. Charge on $C_{4}$:

$Q_{4} = C_{4} \times V = 9 \times 10^{-6} \times 500 = 4.5 \times 10^{-3} \, C$

4. Charge on $C_{1}$:

Voltage across series combination = $500 \, V$

Equivalent capacitance = $3 \, \mu F$

Total charge = $Q = C_{eq} \times V = 3 \times 10^{-6} \times 500 = 1.5 \times 10^{-3} \, C$

In series, all capacitors carry the same charge, so $Q_{1} = Q_{2} = Q_{3} = 1.5 \times 10^{-3} \, C$

Now check statements:

(A) True → $C_{4}$ is in parallel with series of $C_{1}, C_{2}, C_{3}$.

(B) True → Equivalent capacitance = $12 \, \mu F$.

(C) True → Charge on $C_{4} = 4.5 \times 10^{-3} \, C$.

(D) False → $Q_{1} = 1.5 \times 10^{-3} \, C \neq Q_{4} = 4.5 \times 10^{-3} \, C$.

Correct statements: (A), (B), (C)