A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L=5.00 mH and C =3.50 mF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

A capacitor is a device consisting of two conductors in close proximity that are used to store electrical energy. These conductors are insulated from each other. When a capacitor is attached to an AC supply, the resistance produced by it is called capacitive reactance $\left(X_C\right)$.

An inductor is a passive two-terminal device that stores energy in a magnetic field when current passes through it. When an inductor is attached to an AC supply, the resistance produced by it is called inductive reactance $\left(X_L\right)$.

Angular frequency is the measurement of rate of change of waveforms per unit of time. It is generally measure in rad/sec.

Capacitance = C

Inductance = L

Let the reactance of capacitor be X_c and reactance of inductor be X_L, and they can be given by

$X_C=\frac{1}{\omega C},and{X}_L=\omega L$

Where, $\omega$= Angular frequency.

We are asked to find out the angular frequency at which both reactance are same:

$$\begin{gathered} \Rightarrow X_L=X_C \\ \Rightarrow \omega L=\frac{1}{\omega C} \\ \Rightarrow {\omega }^2=\frac{1}{LC} \\ \Rightarrow \omega =\frac{1}{\sqrt{LC}} \end{gathered}$$

Therefore, the angular frequency at which both reactance are same is$\omega =\frac{1}{\sqrt{LC}}$

Given: L=5.00 mH , and $\mathrm{C}=3.50\mathrm{\mu F}$

Angular frequency,

$$\begin{gathered} \omega =\frac{1}{\sqrt{LC}} \\ =\frac{1}{\sqrt{\left(5*{10}^{-3}H\right)*\left(3.{510}^{-6}F\right)}} \\ =7559rad/s \end{gathered}$$

Reactance,

$X_L=X_C=\omega L=(7559rad/s)*(5*{10}^{-3})H)=37.8\Omega$

Therefore, Angular frequency,$\omega =7759rad/sec$ , and Reactance,$X_L=X_C=37.8\Omega$