Question

(a) Compute the reactance of a 0.450-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50 µF capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450 H inductor equal to that of a 2.50 µF capacitor?

Short answer

a) Inductive reactance is 170 Ω when frequency of the source is 60 Hertz, and 1700 Ω when the frequency of the source is 600 Hertz.

b) Capacitive reactance is 1060 Ω when frequency of the source is 60 Hertz, and 160 Ω when the frequency of the source is 600 Hertz.

c) Frequency at which Inductive resistance gets equal to Capacitive resistance is 150 Hertz

Step-by-step solution

Given Data

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 (X_L).

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 (X_C).

Frequency of source, f₁ = 60 Hz, and f₂ = 600 Hz

The inductance of the coil, L = 0.45 H

Capacitance of capacitor, C = 2.50 µF

Determination of Inductive reactance

Inductive reactance is given by,$X_L=2\pi fL$

For f₁ = 60 Hz,

$X_L=2\pi fL=2\pi \left(60Hz\right)(0.45H)=170\Omega$

For f₂ = 600 Hz,

$X_L=2\pi fL=2\pi \left(600Hz\right)(0.45H)=1700\Omega$

Therefore, the Inductive reactance is 170 Ω when frequency of the source is 60 Hertz and 1700 Ω when the frequency of the source is 600 Hertz.

Determination of Capacitive reactance

Capacitive reactance is given by,$X_C=\frac{1}{2\mathrm{\pi fC}}$

For f₁ = 60 Hz,

$X_C=\frac{1}{2\mathrm{\pi fC}}=\frac{1}{2\mathrm{\pi }\left(60\mathrm{Hz}\right)\left(2.5*{10}^{-6}\mathrm{F}\right)}=1060\Omega$

For f₂ = 600 Hz,

$X_C=\frac{1}{2\mathrm{\pi fC}}=\frac{1}{2\mathrm{\pi }\left(600\mathrm{Hz}\right)\left(2.5*{10}^{-6}\mathrm{F}\right)}=106\Omega$

Therefore, the Capacitive reactance is 1060 Ω when frequency of the source is 60 Hertz and 160 Ω when the frequency of the source is 600 Hertz.

Determination of Capacitive reactance

It is given that Inductive resistance is equal to Capacitive resistance, which means,

$$\begin{gathered} X_L=X_C \\ 2\mathrm{\pi fL}=\frac{1}{2\mathrm{\pi fC}} \\ \mathrm{f}=\frac{1}{2\mathrm{\pi }\sqrt{\mathrm{LC}}} \\ \mathrm{f}=\frac{1}{2\mathrm{\pi }\sqrt{\left(0.45\mathrm{H}\right)\left(2.5*{10}^{-6}\mathrm{F}\right)}}=150\mathrm{Hz} \end{gathered}$$

Therefore, the frequency at which Inductive resistance gets equal to Capacitive resistance is 150 Hertz.