Question

The capacitors of capacitance $ 4F $ , $ 6F $ , and $ 12F $ are connected first in series and then in parallel. What is the ratio of equivalent capacitance in the two cases?
(a) $ 2:3 $
(b) $ 11:1 $
(c) $ 1:11 $
(d) $ 1:3 $

Answer 1

Hint: Capacitance is the ability of an electric device, capacitor to store charge. The formula for equivalent capacitance in series and in parallel is to be found. Initially, we find the value of resultant resistance in series and parallel using the given formulas.
The formula for capacitors connected in series is given by, $ \dfrac{1}{{{C_S}}} = \dfrac{1}{{{C_1}}} + \dfrac{1}{{{C_S}}} + \dfrac{1}{{{C_3}}} $
The formula for capacitors connected in parallel is given by, $ {C_P} = {C_1} + {C_2} + {C_3} $ .

Complete answer:
The following data is given to us,
There are three capacitors given to us, let their capacitances be, $ {C_1} = 4F $ , $ {C_2} = 6F $ , $ {C_3} = 12F $ .
Let us consider the first case to be in parallel,
Case 1: when the capacitors are in parallel
 $ {C_P} = {C_1} + {C_2} + {C_3} $
 $ {C_P} = 4 + 6 = 12 = 22F $
The value of equivalent capacitance when they are connected in parallel is $ 22F $
Now let us take the second case where the connection of the capacitors is in series
Case 2: when the capacitors are connected in series,
 $ \dfrac{1}{{{C_S}}} = \dfrac{1}{{{C_1}}} + \dfrac{1}{{{C_S}}} + \dfrac{1}{{{C_3}}} $
 $ \dfrac{1}{{{C_S}}} = \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{{12}} = \dfrac{1}{2} $
We know that this is only the value of the reciprocal of the actual value of the resultant capacitance in series
Taking the reciprocal of this value, we have
 $ {C_S} = 2F $
The question asked is to find the ratio of equivalent capacitance in series to that in parallel connection
 $ \dfrac{{{C_S}}}{{{C_P}}} = \dfrac{{case\,2}}{{case\,1}} = \dfrac{{2F}}{{22F}} = \dfrac{1}{{11}} $
Therefore, we have the ratio of the equivalent resistances to be $ 1:11 $
The right answer is option (C).

Note:
Many might get confused when it comes to the formula because the formula to find the equivalent resistance in series and parallel is the direct opposite to the value of equivalent capacitance.