Question

A capacitance C and an inductance L are operated at the same angular frequency.

Answer 1

A) $\omega = \frac{1}{\sqrt{LC}}$

The magnitude of the capacitive reactance is given by

$X_C = \frac{1}{\omega C}$

where

$\omega$ is the angular frequency

C is the capacitance

While the magnitude of the inductive capacitance is given by

$X_L = \omega L$

where L is the inductance.

Since we want the two reactances to be equal, we have

$X_C = X_L$

So we find

$\frac{1}{\omega C}= \omega L\\\omega^2 = \frac{1}{LC}\\\omega = \frac{1}{\sqrt{LC}}$

B) 7449 rad/s

In this case, we have

$L=5.30 mH = 5.3\cdot 10^{-3}H$ is the inductance

$C= 3.40 \mu F= 3.40 \cdot 10^{-6}F$ is the capacitance

Therefore, substituting in the formula for the angular frequency, we find

$\omega=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{(5.30\cdot 10^{-3}H)(3.40\cdot 10^{-6} F)}}=7449 rad/s$

C) $39.5 \Omega$

Now we can us the formulas of the reactances written in part A). We have:

- Capacitive reactance:

$X_C = \frac{1}{\omega C}=\frac{1}{(7449 rad/s)(3.40\cdot 10^{-6}F)}=39.5 \Omega$

- Inductive reactance:

$X_L = \omega L=(7449 rad/s)(5.30\cdot 10^{-3}H)=39.5 \Omega$