Question

(a) What is the reactance of a 3.00 H inductor at a frequency of 80.0 Hz? (b) What is the inductance of an inductor whose reactance is 120 Ω at 80.0 Hz? (c) What is the reactance of a 4.00 µF capacitor at a frequency of 80.0 Hz? (d) What is the capacitance of a capacitor whose reactance is 120 Ω at 80.0 Hz?

Short answer

a) Inductive reactance is 1510 Ω when frequency of the source is 80 Hertz

b) Inductance is 0.24 H when frequency of the source is 80 Hertz, and reactance is 120 Ω

c) Capacitive reactance is 497 Ω when frequency of the source is 80 Hertz

d) Capacitance is 16.6 µF when frequency of the source is 80 Hertz, and reactance is 120 Ω

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).

The inductance of the coil, L = 3.00 H

Capacitance of capacitor, C = 4.00 µF

Determination of Inductive reactance

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

For frequency, f = 80 Hz,

$X_L=2\pi fL=2\pi \left(80Hz\right)\left(3H\right)=1510\Omega$

Therefore, the Inductive reactance is 1510 Ω when frequency of the source is 80 Hertz.

Determination of Inductance

Inductance of an inductor is given by,$L=\frac{X_L}{2\mathrm{\pi f}}$

For frequency, f = 80 Hz and Reactance X_L = 120 Ω,

$L=\frac{X_L}{2\mathrm{\pi f}}=\frac{120\Omega }{2\mathrm{\pi }\left(80\mathrm{Hz}\right)}=0.24H$

Therefore, the Inductance is 0.24 H when frequency of the source is 80 Hertz, and reactance is 120 Ω.

Determination of Capacitive reactance

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

For frequency, f = 80 Hz,

$X_C=\frac{1}{2\mathrm{\pi fC}}=\frac{1}{2\mathrm{\pi }\left(80\mathrm{Hz}\right)\left(4*{10}^{-6}\mathrm{F}\right)}=497\Omega$

Therefore, the Capacitive reactance is 497 Ω when frequency of the source is 80 Hertz

 Step 5: Determination of Capacitance

Capacitance of a capacitor is given by,$C=\frac{1}{2{\mathrm{\pi fX}}_{\mathrm{C}}}$

For frequency, f = 80 Hz and Reactance X_L = 120 Ω,

$C=\frac{1}{2{\mathrm{\pi fX}}_{\mathrm{C}}}=\frac{1}{2\mathrm{\pi }\left(80\mathrm{Hz}\right)\left(120\mathrm{\Omega }\right)}=16.6\mu F$

Therefore, the Capacitance is 16.6 µF when frequency of the source is 80 Hertz, and reactance is 120 Ω.