Two capacitors of capacitances 8.0\text{ }\mu\text{F} and 4.0\text{ }\mu\text{F} are conne… — JAMB UTME

Two capacitors of capacitances $8.0\text{ }\mu\text{F}$ and $4.0\text{ }\mu\text{F}$ are connected in parallel. This parallel combination is then connected in series with a single $4.0\text{ }\mu\text{F}$ capacitor across a $36.0\text{ V}$ d.c. voltage source. What is the potential difference across the single $4.0\text{ }\mu\text{F}$ capacitor?

A
$9.0\text{ V}$
B
$14.4\text{ V}$
$27.0\text{ V}$ Answer
D
$36.0\text{ V}$

Answer

The potential difference across the single

4.0\text{ }\mu\text{F}

capacitor is

27.0\text{ V}

The two parallel capacitors (

8.0\text{ }\mu\text{F}

and

4.0\text{ }\mu\text{F}

) combine directly to give an equivalent capacitance of

12.0\text{ }\mu\text{F}

. This equivalent capacitor is in series with the single

4.0\text{ }\mu\text{F}

capacitor across the

36.0\text{ V}

source. Using the voltage divider rule for series capacitors, the voltage across the single

4.0\text{ }\mu\text{F}

capacitor is

V = 36.0 \times \frac{12.0}{4.0 + 12.0} = 27.0\text{ V}

Step-by-Step Solution

Calculate the equivalent capacitance of the two parallel capacitors.

C_p = 8.0\text{ }\mu\text{F} + 4.0\text{ }\mu\text{F} = 12.0\text{ }\mu\text{F}

Capacitors in parallel add algebraically.

Calculate the total equivalent capacitance of the entire circuit.

C_T = \frac{4.0 \times 12.0}{4.0 + 12.0} = \frac{48.0}{16.0} = 3.0\text{ }\mu\text{F}

The single

4.0\text{ }\mu\text{F}

capacitor and the

12.0\text{ }\mu\text{F}

parallel combination are in series.

Find the total charge supplied by the battery.

Q = C_T \times V = 3.0\text{ }\mu\text{F} \times 36.0\text{ V} = 108.0\text{ }\mu\text{C}

The total charge is equal to the total capacitance multiplied by the total voltage.

Calculate the potential difference across the single

4.0\text{ }\mu\text{F}

capacitor.

V_1 = \frac{Q}{C_1} = \frac{108.0\text{ }\mu\text{C}}{4.0\text{ }\mu\text{F}} = 27.0\text{ V}

In a series connection, the charge on the single capacitor equals the total charge.

Key Concept

Potential difference distribution in mixed series-parallel capacitor networks

Estimated Time:1m 30s

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