Question

When an additional charge of $2\;C$ is given to a capacitor , energy stored in it is increased by $21\%$. The original charge of the capacitor is:

& A.30C \\\ & B.40C \\\ & C.10C \\\ & D.20C \\\ \end{aligned}$$

Answer 1

We know that the energy of the capacitor is $E=\dfrac{1}{2}CV^{2}$. Since here, it is given that the energy of the capacitance increases by some fraction, when the charge is increased, we can take the ratio of the new energy with the old energy to find the charge of the capacitor.

Formula used: $E=\dfrac{1}{2}CV^{2}$.

Complete step by step answer:
A capacitor is a two terminal component that stores electrical energy in the form of potential energy, and later discharges them. This property is called the capacitance of the capacitor.
Let the charge $Q$ produced when a voltage $V$ is applied to the capacitance$C$, then$Q=CV$.
Let us assume that initial charge be $Q$, then the energy produced is given as $E=\dfrac{Q^{2}}{2C}$
Let the new charge be given as $Q\prime=Q+2$
And the new energy be given as $E\prime=1.21E$
Then, $1.21E=\dfrac{(Q+2)^{2}}{2C}$
Taking the ratio of the energy, we get,
$\dfrac{1.21E}{E}=\dfrac{\dfrac{(Q+C)^{2}}{2C}}{\dfrac{Q^{2}}{2C}}$
$\implies 1.21=\dfrac{(Q+C)^{2}}{Q^{2}}$
$\implies 1.1=\dfrac{Q+C}{Q}$
$\implies 1.1Q-Q=C$
$\implies 0.1Q=C$
Given that $C=2$
$\implies Q=20C$

So, the correct answer is “Option D”.

Additional Information: A capacitor can store electrical energy, and behaves as a temporary battery. They are used mainly to maintain the power supply while batteries are being changed. It can also store information in the form of binary digits. It is the main component used in full wave and half wave rectifiers. (symbol: F), named after the English physicist Michael Faraday. A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates. The series of capacitors is the sum of reciprocal of its individual capacitors, whereas in resistance the parallel is the sum of reciprocal of its individual resistors. Also remember that capacitors can charge and discharge.

Note: The series of capacitor is the sum of reciprocal of its individual capacitors, whereas in resistance the parallel is the sum of reciprocal of its individual resistors. Also remember that capacitors can charge and discharge.