Seven capacitors, each of capacitance \( 2\,\mu\mathrm{F} \), are to be connected to obtain an effective capacitance of \( \frac{10}{11}\,\mu\mathrm{F} \). Which of the following combinations will do this?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → the diagram in Option (1)
Capacitors in parallel add directly, \( C_{\mathrm{p}}=C_1+C_2+\ldots \). For capacitors in series, the reciprocals add, \( 1/C_{\mathrm{s}}=1/C_1+1/C_2+\ldots \). In each option, a parallel group of \( 2\,\mu\mathrm{F} \) capacitors is in series with a chain of \( 2\,\mu\mathrm{F} \) capacitors. All capacitances below are in \( \mu\mathrm{F} \).
Option (1) — Correct. From the figure, five capacitors in parallel give \( 10\,\mu\mathrm{F} \), in series with two more.
\( \begin{aligned} \frac{1}{C} &= \frac{1}{10}+\frac{1}{2}+\frac{1}{2} \\ &= \frac{11}{10} \end{aligned} \)
\( \displaystyle C=\frac{10}{11}\,\mu\mathrm{F} \)
Option (2) — Incorrect. From the figure, four capacitors in parallel give \( 8\,\mu\mathrm{F} \), in series with three more.
\( \begin{aligned} \frac{1}{C} &= \frac{1}{8}+\frac{3}{2} \\ &= \frac{13}{8} \end{aligned} \)
\( \displaystyle C=\frac{8}{13}\,\mu\mathrm{F} \)
Option (3) — Incorrect. From the figure, three capacitors in parallel give \( 6\,\mu\mathrm{F} \), in series with four more.
\( \begin{aligned} \frac{1}{C} &= \frac{1}{6}+\frac{4}{2} \\ &= \frac{13}{6} \end{aligned} \)
\( \displaystyle C=\frac{6}{13}\,\mu\mathrm{F} \)
Option (4) — Incorrect. From the figure, two capacitors in parallel give \( 4\,\mu\mathrm{F} \), in series with five more.
\( \begin{aligned} \frac{1}{C} &= \frac{1}{4}+\frac{5}{2} \\ &= \frac{11}{4} \end{aligned} \)
\( \displaystyle C=\frac{4}{11}\,\mu\mathrm{F} \)