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Q.
Five capacitors, each of capacitance value $C$ are connected as shown in the figure. The ratio of capacitance between $P$ and $R$, and the capacitance between $P$ and $Q$, is :
AIIMSAIIMS 2006Electrostatic Potential and Capacitance
Solution:
The capacitors having same charge through them will be in series combination.
If a source is connected between points $P$ and $R$, same charge will flow through two capacitors in arms $P Q$ and $Q R$. Similarly, same charge will flow through capacitors in arms PT, TS and SR.
So, equivalent capacitance of left side
$C '=\frac{ C \times C }{ C + C }=\frac{ C }{2}$
and equivalent capacitance of right side
$\frac{1}{C''} =\frac{1}{C}+\frac{1}{C}+\frac{1}{C}=\frac{3}{C} $
$\therefore C''=\frac{C}{3}$
Now, $C'$ and $C' '$ will in parallel combination, hence,
$C _{1}= C'+ C''=\frac{ C }{2}+\frac{ C }{3}=\frac{5 C }{6}$
Similarly, if a source is connected between points $P$ and $Q$, then equivalent capacitance
$C_{2} =C+\frac{C}{4} $
$=\frac{5 C}{4}$
Hence, the required ratio is given by
$\frac{ C _{1}}{ C _{2}}=\frac{5 C / 6}{5 C / 4}=\frac{2}{3}$
