Capacitors: Exam-Pattern Drill (1)

easy • 2 min read
Tags Capacitors

Question

Three capacitors of capacitance 2 μF2\,\mu\text{F}, 3 μF3\,\mu\text{F} and 6 μF6\,\mu\text{F} are connected first in series and then in parallel across a 12 V12\,\text{V} battery. Find the equivalent capacitance, total charge stored, and total energy stored in each configuration.

Solution — Step by Step

For series capacitors, reciprocals add:

1Cs=12+13+16=3+2+16=1\frac{1}{C_s} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3 + 2 + 1}{6} = 1

Cs=1 μFC_s = 1\,\mu\text{F}

In series, the same charge sits on every capacitor:

Qs=CsV=1×10−6×12=12 μCQ_s = C_s V = 1 \times 10^{-6} \times 12 = 12\,\mu\text{C}

Energy:

Us=12CsV2=12×10−6×144=72 μJU_s = \frac{1}{2}C_s V^2 = \frac{1}{2} \times 10^{-6} \times 144 = 72\,\mu\text{J}

For parallel capacitors, capacitances add:

Cp=2+3+6=11 μFC_p = 2 + 3 + 6 = 11\,\mu\text{F}

In parallel, the voltage across each capacitor is the same (12 V):

Qp=CpV=11×12=132 μCQ_p = C_p V = 11 \times 12 = 132\,\mu\text{C}

Up=12×11×10−6×144=792 μJU_p = \frac{1}{2} \times 11 \times 10^{-6} \times 144 = 792\,\mu\text{J}

Final: Series Cs=1 μC_s = 1\,\muF, Q=12 μQ = 12\,\muC, U=72 μU = 72\,\muJ. Parallel Cp=11 μC_p = 11\,\muF, Q=132 μQ = 132\,\muC, U=792 μU = 792\,\muJ.

Why This Works

Series capacitors store less total charge than any individual capacitor because the equivalent capacitance is smaller than the smallest member. Parallel capacitors store more total charge because the equivalent capacitance is the sum.

The energy ratio Up/Us=11U_p/U_s = 11 matches Cp/CsC_p/C_s exactly — at fixed voltage, energy scales with capacitance.

Alternative Method

For the series energy, sum the individual energies. Each capacitor has the same charge 12 μ12\,\muC, so:

Ui=Q22CiU_i = \frac{Q^2}{2C_i}

Utotal=(12)22(12+13+16)=72×1=72 μJU_{\text{total}} = \frac{(12)^2}{2}\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{6}\right) = 72 \times 1 = 72\,\mu\text{J}

Same answer.

Common Mistake

Students apply the resistor formula to capacitors — series capacitors add reciprocally, not directly. The rule is “opposite of resistors”: series reciprocals add for capacitors, parallel values add for capacitors. Easy to swap under exam pressure.

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