Question

A capacitance $C$ is required to make the power factor unity in an ac circuit \[\left( {v = 50Hz} \right)\]. The circuit contains an inductance of \[0.5H\]and a resistance of \[15\Omega \]. The value of $C$ should be-
A. \[20\mu F\]
B. \[10\mu F\]
C. \[2\mu F\]
D. \[1\mu F\]

Answer 1

Hint: As we know that the power factor of an AC circuit is defined as the ratio of the real power consumed by a circuit to the apparent power consumed by the same circuit. Here power factor unity is given. By using this, we will find one condition in which inductive and capacitive reactance both are equal.

Complete step by step answer:
According to question, power factor is unity which means \[\operatorname{Cos} \phi = 1\]
\[\phi = 0\] . Now inductive and capacitive reactance both are equal.
\[{X_L} = {X_C}\]
We know value for \[{X_L}\& {X_C}\], substitute the value of both, we get-
\[\omega L = \dfrac{1}{{\omega C}}\] -- (1)
From equation (1),
\[C = \dfrac{1}{{{\omega ^2}L}}\]
Now we know value of ω=\[\sqrt {2\pi f} \], substitute it in the resulted equation, we get-
\[ \omega= \dfrac{1}{{2\pi f \times 0.5}}\]
\[\Rightarrow C = \dfrac{1}{{314 \times 0.5}}\]
\[\therefore C = 20\mu F\]

So, the value of capacitance, \[C = 20\mu F\]. Hence, Option A is correct.

Note: Reactance is just the opposition that is offered by capacitor and inductor in any circuit to flow the AC current in the circuit. It is of two types- Capacitive reactance ${X_C}$ and inductive reactance ${X_L}$. Inductive reactance is the opposition or obstacle offered by the inductor in an AC circuit to the flow of AC current whereas the capacitive reactance is inversely proportional to the capacitance and the single frequency.