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Physics
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Capacitance

NEET · Physics · Capacitance

A circuit consists of three capacitors C_1 = 2\ , C_2 = 3\ , and...

A circuit consists of three capacitors C1=2 μFC_1 = 2\ \mu\text{F}, C2=3 μFC_2 = 3\ \mu\text{F}, and C3=5 μFC_3 = 5\ \mu\text{F} connected in parallel. This entire parallel combination is connected in series with a fourth capacitor C4=10 μFC_4 = 10\ \mu\text{F} across an unknown DC voltage source VV. If the charge on capacitor C1C_1 is 12 μC12\ \mu\text{C}, the total voltage VV of the source and the charge on capacitor C4C_4 respectively are:

Options

  1. A12 V12\text{ V} and 60 μC60\ \mu\text{C}
  2. B6 V6\text{ V} and 60 μC60\ \mu\text{C}
  3. C7.2 V7.2\text{ V} and 12 μC12\ \mu\text{C}
  4. D12 V12\text{ V} and 12 μC12\ \mu\text{C}

Correct answer

A. 12 V12\text{ V} and 60 μC60\ \mu\text{C}

Step-by-step solution

The capacitors C1,C2C_1, C_2, and C3C_3 are connected in parallel. The voltage across this parallel combination is the same for each capacitor.
Voltage across the parallel group, Vp=Q1C1=12 μC2 μF=6 VV_p = \dfrac{Q_1}{C_1} = \dfrac{12\ \mu\text{C}}{2\ \mu\text{F}} = 6\text{ V}.

The equivalent capacitance of the parallel group is:
Cp=C1+C2+C3=2 μF+3 μF+5 μF=10 μFC_p = C_1 + C_2 + C_3 = 2\ \mu\text{F} + 3\ \mu\text{F} + 5\ \mu\text{F} = 10\ \mu\text{F}.

The total charge on the parallel group is:
Qp=Cp×Vp=10 μF×6 V=60 μCQ_p = C_p \times V_p = 10\ \mu\text{F} \times 6\text{ V} = 60\ \mu\text{C}.

Since capacitor C4C_4 is connected in series with the parallel group, it carries the same total charge as the parallel group.
Therefore, the charge on C4C_4 is Q4=60 μCQ_4 = 60\ \mu\text{C}.

The voltage across C4C_4 is:
V4=Q4C4=60 μC10 μF=6 VV_4 = \dfrac{Q_4}{C_4} = \dfrac{60\ \mu\text{C}}{10\ \mu\text{F}} = 6\text{ V}.

The total voltage of the source VV is the sum of the voltages across the parallel group and C4C_4:
V=Vp+V4=6 V+6 V=12 VV = V_p + V_4 = 6\text{ V} + 6\text{ V} = 12\text{ V}.

Answer: 12 V12\text{ V} and 60 μC60\ \mu\text{C}
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