Question

Find the charges on the four capacitors of capacitances 1 μF, 2 μF, 3 μF and 4 μF shown in the figure (32-E32).

Answer 1

When the capacitors are fully charged, they attain steady state and no current flows through them. Then, equivalent resistance of the circuit,
$R_{\mathrm{eff}}=\frac{3\times 6}{3+6}=2\mathrm{\Omega }$
Current through the circuit,
$i=\frac{6}{2}=3\mathrm{A}$
Current i is divided in the inverse ratio of the resistance in each branch. One branch has resistance of 3 Ω and the other branch has resistance of 6 Ω.
Current i' through the 3 Ω branch,
$i\text{'}=\frac{6}{9}i=\frac{2}{3}\times 3\mathrm{A}=2\mathrm{A}$
Current i'' through the 6 Ω branch,
$i\text{'}\text{'}=i-i\text{'}=1\mathrm{A}$

Voltage across the 1 Ω resistor = 2 A × 1 Ω = 2 V
Charge on the 1 μF capacitor = 2 × 1 μF = 2 μC

Voltage across the 2 Ω resistor = 2 Ω × 2 A = 4 V
Charge on the 2 μF capacitor = 4V × 2 μF = 8 μC

Voltage across each 3 Ω resistor = 3 Ω × 1 A = 3 V
Charge on the 4 μF capacitor = 3 × 4 μC = 12 μC

Charge on the 3 μF capacitor = 3 × 3 μC = 9 μC