Question 440: Electrostats & Capacitance

Question

Five capacitors, each of capacitance ' $C$ ' are connected as shown in the figure. The ratio of equivalent capacitance between P and R and the equivalent capacitance between P and Q is

Accepted Answer

$$ 	ext { Let the capacitors be } \mathrm{C}_1, \mathrm{C}_2, \mathrm{C}_3, \mathrm{C}_4 	ext { and } \mathrm{C}_5 $$
Between P and Q- $C_2, C_3, C_4$ and $C_5$ are connected in series and $\mathrm{C}_1$ is connected in parallel with them.
For the series combination
$$ \frac{1}{C_{	ext {series }}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{4}{C} $$
Equivalent capacitance,
$$ C_{P Q}=C_{	ext {series }}+C=\frac{C}{4}+C=\frac{5 C}{4} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)$$
Between points P and $\mathrm{R}-\mathrm{C}_2, \mathrm{C}_3$ and $\mathrm{C}_4$ are connected in series and $C_1$ and $C_5$ connected in series are connected in parallel with them
Between points P and R , capacitors $\mathrm{C}_2, \mathrm{C}_3$, and $\mathrm{C}_4$ are connected in series, and this series combination is in parallel with another series combination of capacitors $\mathrm{C}_1$ and $\mathrm{C}_5$
$$ \begin{aligned} & \frac{1}{C_{	ext {series }}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{3}{C} \Rightarrow C_{	ext {series } 1}=\frac{C}{3} \ & \frac{1}{C_{	ext {series } 2}}=\frac{1}{C}+\frac{1}{C}=\frac{2}{C} \Rightarrow C_{	ext {series } 2}=\frac{C}{2} \end{aligned} $$
$$ \begin{aligned} &C_{P R}=C_{	ext {series } 1}+C_{	ext {series } 2}=\frac{C}{3}+\frac{C}{2}=\frac{5 C}{6} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)\ &	ext { From (i) and (ii), }\ &\frac{C_{P R}}{C_{P Q}}=\frac{5 C / 6}{5 C / 4}=\frac{2}{3} \end{aligned} $$

Answer

$1: 4$

Answer

$2: 3$

Answer

$3: 1$

Answer

$5: 2$

Electrostats & Capacitance

Five capacitors, each of capacitance ' C ' are connected as shown in the figure. The ratio of equivalent capacitance between P and R and the equivalent capacitance between P and Q is

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