Question

he resonant frequency of the L-C circuit is,
(A) \[\dfrac{1}{2\pi \sqrt{LC}}\]
(B) \[\dfrac{1}{2\pi }\sqrt{\dfrac{L}{C}}\]
(C) \[\dfrac{1}{4\pi }\sqrt{\dfrac{L}{C}}\]
(D) \[\dfrac{1}{2\pi }\sqrt{\dfrac{C}{L}}\]

Answer 1

Hint: In the L-C circuit the resonance occurs when the capacitive and inductive reactance are equal in magnitude because in the L-C circuit inductive reactance increases with the time and the capacitive reactance decreases with the time. At an instance time both the values become the same so the resonance occurs in the circuit.

Complete answer:
In this L-C circuit inductive reactance \[{{X}_{L}}=\omega L\]and capacitive reactance is\[{{X}_{C}}=\dfrac{1}{\omega C}\].
The current in the L-C-R circuit is given by \[I=\dfrac{{{V}_{m}}\cos (\omega t-\delta )}{\left| Z \right|}\]
And maximum current is given by \[I=\dfrac{{{V}_{m}}}{\left| Z \right|}=\dfrac{{{V}_{m}}}{\sqrt{{{R}^{2}}+{{(\omega L-\dfrac{1}{\omega C})}^{2}}}}\]
But in L-C circuit the resistance is absent then the current is \[I=\dfrac{{{V}_{m}}}{\left| Z \right|}=\dfrac{{{V}_{m}}}{\sqrt{{{(\omega L-\dfrac{1}{\omega C})}^{2}}}}\]
Now the change in the value of \[\omega \]leads towards the change in the current flowing in the circuit and for a particular value of \[\omega ={{\omega }_{0}}\] the current becomes maximum in the circuit. The denominator must tend to zero for the maximum current.

For that inductive reactance and capacitive reactance must be equal.
\[{{X}_{L}}={{X}_{C}}\]
\[\therefore {{\omega }_{0}}L=\dfrac{1}{{{\omega }_{0}}C}\]
\[\therefore {{\omega }_{0}}^{2}=\dfrac{1}{LC}\]
\[\therefore {{\omega }_{0}}=\dfrac{1}{\sqrt{LC}}\]
But\[{{\omega }_{0}}=2\pi {{f}_{0}}\],
\[\therefore 2\pi {{f}_{0}}=\dfrac{1}{\sqrt{LC}}\]
And \[\therefore {{f}_{0}}=\dfrac{1}{2\pi \sqrt{LC}}\]
Here \[{{\omega }_{0}}\] is called angular resonant frequency and \[{{f}_{0}}\] is called resonant frequency.

So, the correct answer is “Option A”.

Note:
Since \[{{X}_{L}}={{X}_{C}}\], Therefore, there is absolutely no reactive component to the total impedance at the resonant frequency. In the absence of any resistance, the current rises without limit and becomes theoretically infinite and the voltage source behaves like an almost short-circuit.

The real-world problem is that no circuit is complete without resistance, and the resistance present will serve to limit the current from the source. But for very small resistance it does not affect because the current remains high for very small resistance.