Question 599: Electrostatics & Capacitance

Question

Which of the following combinations of 7 identical capacitors each of $2 \mu \mathrm{~F}$ gives a resultant capacitance of $10 / 11 \mu \mathrm{~F}$ ?

Accepted Answer

For $n$ capacitors i.e., $C_4, C_2, \ldots \ldots \ldots . C_n$ their equivalent capacitance, when connected in
(a) series is given as, $\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}+\ldots \ldots$
and
(b) parallel is given as, $C_p=C_1+C_2+\ldots \ldots \ldots$
In the given problem, we will check the net capacitance in each option and compare it with the desired capacitance mentioned in it.
(a) 3 capacitors are in parallel and 4 capacitors are connect in series, i.e.,
where, $C_{A B}$ is the equivalent capacitance of parallel branches between $A$ and $B$ and $C_{B D}$ is equivalent capacitance of series capacitors between $B$ and $D$. So, equivalent capacitance.
$$C_{	ext {eq }}=\frac{C_{A B} \cdot C_{B D}}{C_{A B}+C_{B D}}=\frac{6 	imes \frac{2}{4}}{6+\frac{2}{4}}=\frac{6}{13} \mu \mathrm{~F}$$
Similarly, (b) 2 capacitors are in parallel and 5 capacitors are connect in series i.e.,
So, equivalent capacitance,
$$C_{	ext {eq }}=\frac{4 	imes \frac{2}{5}}{4+\frac{2}{5}}=\frac{4}{11} \mu \mathrm{~F}$$
(c) 4 capacitors are in parallel and 3 capacitors are connect in series i.e.,
So, equivalent capacitance,
$$C_{	ext {eq }}=\frac{8 	imes \frac{2}{3}}{8+\frac{2}{3}}=\frac{16}{26}=\frac{8}{13} \mu \mathrm{~F}$$
(d) 5 capacitors are in parallel and 2 capacitors are connect in series i.e.,
So, equivalent capacitance,
$$C_{e q}=\frac{10 	imes 1}{10+1}=\frac{10}{11} \mu \mathrm{~F}$$
So, the correct option is (d).

Answer

3 in parallel and 4 in series

Answer

2 in parallel and 5 in series

Answer

4 in parallel and 3 in series

Answer

5 in parallel and 2 in series

Electrostatics & Capacitance

Which of the following combinations of 7 identical capacitors each of 2 \mu \mathrm{~F} gives a resultant capacitance of 10 / 11 \mu \mathrm{~F} ?

Choose the correct answer: