Question

The natural frequency of the circuit shown in adjoining figure is

(A)$\dfrac {1} {2\pi \sqrt{LC}}$
(B)$\dfrac {1} {2\pi \sqrt{2LC}}$
(C)$\dfrac {2} {2\pi \sqrt{LC}}$
(D)Zero

Answer 1

The above question is based on the theory of LC oscillator. Firstly, we will do a calculation for the capacitor and find the equivalent value of capacitor for the given figure. Then, we will do the calculation for the inductor, here we will calculate the equivalent value of the inductor for the given figure. Now, we will put these values in the expression of natural frequency of the circuit and get the result.

Complete step by step answer:
As we can seen from the figure that two capacitor are in series so, by applying series formula of capacitor:
\begin {align} & {{L} _{s}} = {{L} _ {1}} + {{L} _ {2}} \\\ & \Rightarrow {{L} _{s}} =L+L \\\ & \Rightarrow {{L} _{s}} =2L \\\ \end{aligned}
Now, we will do calculation for capacitor, we can see that here capacitor is connected in series so, by applying series formula of capacitor we can write:
\begin {align} & \dfrac{1}{{{C}_{s}}}=\dfrac{1}{{{C}_{1}}}+\dfrac{1}{{{C}_{2}}} \\\ & \Rightarrow \dfrac{1}{{{C}_{s}}}=\dfrac{1}{C}+\dfrac{1}{C} \\\ & \Rightarrow \dfrac{1}{{{C}_{s}}}=\dfrac{2}{C} \\\ & \Rightarrow {{C} _{s}} =\dfrac{C}{2} \\\ \end{aligned}
For the natural frequency of the given LC circuit we have the expression as,
$v=\dfrac {1} {2\pi \sqrt{{{L}_{s}}{{C}_{s}}}}$
\begin {align} & \Rightarrow v=\dfrac {1} {2\pi \sqrt{2L}\times \sqrt{\dfrac{C}{2}}} \\\ & \therefore v=\dfrac {1} {2\pi \sqrt{LC}} \\\ & \\\ \end{aligned}
So, the value of natural frequency for the given figure will be$\dfrac {1} {2\pi \sqrt{LC}}$.

So, the correct answer is “Option A”.

Note: The most important thing we must focus on is whether the capacitors and inductors are connected in parallel or series. This analysis will help you in solving complex circuits containing capacitors and inductors. We must remember the formula of resistor and from this formula only we can learn the inductor and capacitors series and parallel connection formula. For series connection of capacitor, the formula is similar to the parallel connection of resistor whereas, for series connection of inductor, the formula is similar to the series connection of resistor. In series, all the values of resistor are added up together whereas, in parallel connection of resistor, the reciprocal of all the values of resistor is added to get equivalent resistor.