Case: Read the passage and answer the question.
A capacitor is a system of two conductors separated by an insulator. The electric field in the region between the conductors is proportional to the charge Q. That is, if the charge on the capacitor is, say, doubled, the electric field will also be doubled at every point. A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance. Dielectrics are introduced between the plates to change the nature of capacitors. Capacitors can be connected in series and in parallel in a circuit.
Three capacitors of capacitance \( 5\,\mu\mathrm{F} \), \( 10\,\mu\mathrm{F} \) and \( 15\,\mu\mathrm{F} \) are connected in series. What is the equivalent capacitance?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \frac{30}{11}\,\mu\mathrm{F} \)
Capacitors in series carry the same charge, and their potential differences add. So the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances.
For the \( 5\,\mu\mathrm{F} \), \( 10\,\mu\mathrm{F} \) and \( 15\,\mu\mathrm{F} \) capacitors (values in \( \mu\mathrm{F} \)):
\( \begin{aligned} \frac{1}{C} &= \frac{1}{5}+\frac{1}{10}+\frac{1}{15} \\ &= \frac{6+3+2}{30} \\ &= \frac{11}{30}\,\mu\mathrm{F}^{-1} \end{aligned} \)
\( \displaystyle C=\frac{30}{11}\,\mu\mathrm{F}\approx2.7\,\mu\mathrm{F} \)
The series value is smaller than the smallest capacitance, \( 5\,\mu\mathrm{F} \).