Two capacitors of capacitances C_1 = 6\text{ }\mu\text{F} and C_2 = 12\text{ }\mu\text{F} … — JAMB UTME

Two capacitors of capacitances $C_1 = 6\text{ }\mu\text{F}$ and $C_2 = 12\text{ }\mu\text{F}$ are connected in series across a $180\text{ V}$ direct current power supply. After the capacitors are fully charged, the supply is disconnected. A dielectric material of dielectric constant $K = 4$ is then inserted to completely fill the space between the plates of $C_1$ . What is the new potential difference across $C_1$ ?

$30\text{ V}$ Answer
B
$60\text{ V}$
C
$135\text{ V}$
D
$15\text{ V}$

Answer

30\text{ V}

The initial equivalent capacitance of the series combination is

4\text{ }\mu\text{F}

, which charges each capacitor to

720\text{ }\mu\text{C}

. Because the circuit is disconnected from the battery, charge is conserved. Inserting a dielectric of constant

K = 4

increases

C_1

24\text{ }\mu\text{F}

, resulting in a potential difference of

V = \frac{Q}{C_1'} = \frac{720\text{ }\mu\text{C}}{24\text{ }\mu\text{F}} = 30\text{ V}

Step-by-Step Solution

Calculate the initial equivalent capacitance of the series network

C_{eq} = 4\text{ }\mu\text{F}

Capacitors in series combine reciprocally:

\frac{1}{C_{eq}} = \frac{1}{6} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}\text{ }\mu\text{F}^{-1}

Determine the charge stored on each capacitor prior to disconnection

Q = 720\text{ }\mu\text{C}

Total charge supplied by the

180\text{ V}

battery is

Q = C_{eq} V = 4\text{ }\mu\text{F} \times 180\text{ V} = 720\text{ }\mu\text{C}

. In series, each capacitor holds this same charge.

Calculate the modified capacitance of

C_1

after dielectric insertion

C_1' = 24\text{ }\mu\text{F}

A dielectric of constant

K = 4

scales capacitance by

K

C_1' = K \times C_1 = 4 \times 6\text{ }\mu\text{F} = 24\text{ }\mu\text{F}

Calculate the final potential difference across

C_1

using charge conservation

V_1' = 30\text{ V}

Because the source is disconnected, charge

Q = 720\text{ }\mu\text{C}

C_1

remains constant. Therefore,

V_1' = \frac{Q}{C_1'} = \frac{720\text{ }\mu\text{C}}{24\text{ }\mu\text{F}} = 30\text{ V}

Key Concept

Effect of dielectrics and charge conservation in disconnected series capacitor circuits

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