Question:

A series combination of $n_1$ capacitors, each of capacity $C_1$ is charged by source of potential difference $4\, V$. When another parallel combination of $n_2$ capacitors each of capacity $C_2$ is charged by a source of potential difference $V$, it has the same total energy stored in it as the first combination has. The value of $C_2$ in terms of $C_1$ is then

Updated On: Jul 31, 2023
  • $16 \frac{n_{2}}{n_{1}}C_{1}$
  • $ \frac{2 C_{1}}{n_{1}n_{2}}$
  • $2 \frac{n_{1}}{n_{2}}C_{2}$
  • $ \frac{16 C_{1}}{n_{1}n_{2}}$
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The Correct Option is D

Solution and Explanation

Equivalent capacitance of $n_2$ number of capacitors each of capacitance $C_2$ in parallel $= n_2C_2$ Equivalent capacitance of $n_1$ number of capacitors each of capacitances $C_1$ in series. Capacitance of each is $C_{1} \frac{C_{1}}{n_{1}}$ According to question, total energy stored in both the combinations are same $i.e, \frac{1}{2}\left(\frac{C_{1}}{n_{1}}\right)\left(4V\right)^{2} = \frac{1}{2}\left(n_{2}C_{2}\quad\right)^{2}$ $\therefore\,C_{2} = \frac{16C_{1}}{n_{1}n_{2}}$

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Concepts Used:

Electrostatic Potential and Capacitance

Electrostatic Potential

The potential of a point is defined as the work done per unit charge that results in bringing a charge from infinity to a certain point.

Some major things that we should know about electric potential:

  • They are denoted by V and are a scalar quantity.
  • It is measured in volts.

Capacitance

The ability of a capacitor of holding the energy in form of an electric charge is defined as capacitance. Similarly, we can also say that capacitance is the storing ability of capacitors, and the unit in which they are measured is “farads”.

Read More: Electrostatic Potential and Capacitance

The capacitor is in Series and in Parallel as defined below;

In Series

Both the Capacitors C1 and C2 can easily get connected in series. When the capacitors are connected in series then the total capacitance that is Ctotal is less than any one of the capacitor’s capacitance.

In Parallel

Both Capacitor C1 and C2 are connected in parallel. When the capacitors are connected parallelly then the total capacitance that is Ctotal is any one of the capacitor’s capacitance.