Question

An inductor of 100 mH is connected in series with a resistance, a variable capacitance and an AC source of frequency 2.0 kHz; The value of the capacitance so that maximum current may be drawn into the circuit.
A. 50 nf
B. 60 nf
C. 63 nF
D. 79 nF

Answer 1

Hint:The inductive reactance tells how much the inductor possesses an ac signal. In series LCR circuit. It consists of an inductor(L), capacitor(C) and a resistor(R). Depending upon the values of\[{X_L}{\rm{ and }}{{\rm{X}}_C}\]we can find capacitance of a circuit.

Formula used:
\[{X_L} = \omega L\]
Where, \[{X_L}\] is inductive reactance and L is the inductor.
\[{X_C} = \dfrac{1}{{\omega C}}\]
Where, \[{X_C}\] is capacitive reactance and C is capacitance.

Complete step by step solution:
Given Frequency is ,
\[\begin{array}{l}\upsilon = 2\,kHz{\rm{ }} = 2 \times {10^3}Hz\end{array}\]
Inductance,
\[\begin{array}{l}L = 100\,mH{\rm{ = 100}} \times {10^{ - 3}}H\end{array}\]
Now we can calculate angular frequency as,
\[\omega = 2\pi \upsilon \\
\Rightarrow \omega = 2\pi \times 2 \times {10^3}\\
\Rightarrow \omega = 4\pi \times {10^3}\]

In series LCR circuit
\[X = {X_C} - {X_L}\]
For maximum current reactance, $X$ must be 0.
\[0 = {X_C} - {X_L}\]
Hence, we get
\[{X_L} = {X_C}\]
Therefore,
\[\omega L = \dfrac{1}{{\omega C}}\]
\[\Rightarrow C = \dfrac{1}{{{\omega ^2}L}}\]

Substituting the given values
\[C = \dfrac{1}{{{{(4\pi \times {{10}^3})}^2} \times 100 \times {{10}^{ - 3}}}}\]
\[\begin{array}{l} \Rightarrow C= 62.5 \times {10^{ - 9}}F\\ \Rightarrow C= 62.5nF\\ \therefore C = 63\,nF\end{array}\]
Therefore the value of the capacitance so that maximum current may be drawn into the circuit is 63nF.

Hence option C is the correct answer.

Note: An LCR circuit is also known as an RLC circuit, resonant circuit or tuned circuit. It is an electrical circuit consisting of an inductor, resistor and capacitor. LCR circuit can be more understood by the term phasor. A rotating quantity is known as a phasor. In an LCR circuit using current across the circuit is to be our reference phasor due to it remaining the same for all the components in a series LCR circuit. A capacitor consists of two-terminal electrical devices that can be stored in the form of an electric charge.