A 150Ω resistor is connected in series with a 0.250H inductor and an ac source. The voltage across the resistor is v_R = (3.80 V) cos [(720 rad/s) t]. (a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage v_L across the inductor.

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).

Resistance is measure of opposition to the flow of current in a closed electrical circuit. It is measured in Ohm (Ω).

Series Resistance, R = 150 Ω

The inductance of the coil, L = 0.250 H

Equation of voltage across resistor is

$v_R=(3.80V)cos\left[\right(720rad/s\left)t\right]$

By using ohm’s law

$$\begin{gathered} \mathrm{l}=\frac{{\mathrm{v}}_{\mathrm{R}}}{\mathrm{R}} \\ =\left(\frac{3.80\mathrm{V}}{150\mathrm{\Omega }}\right)\cos \left[\left(720\mathrm{rad}/\mathrm{s}\right)\mathrm{t}\right] \\ =\left(0.025\mathrm{A}\right)\cos \left[\left(720\mathrm{rad}/\mathrm{s}\right)\mathrm{t}\right] \end{gathered}$$

Therefore, the expression of circuit current is given by$I=(0.025A)\cos [(720rad/s)t]$

Inductive reactance is given by,

$$\begin{gathered} X_L=\omega L \\ =\left(720rad/s\right)\left(0.250H\right) \\ =180\Omega \end{gathered}$$

Therefore, the inductive reactance of the inductor is 180 Ω.

Voltage across inductor leads the current by a phase different of – 90 degrees, so the expression of voltage can be given as –

$$\begin{gathered} v_L=v_L\cos \left(\omega t+90°\right) \\ =I{X}_L\cos \left(\omega t+90°\right) \\ =\left(0.025A\right)\left(180\Omega \right)\cos \left[\left(720rad/s\right)t+90°\right] \\ =-\left(4.56V\right)\sin \left[\left(720rad/s\right)t\right] \end{gathered}$$

Therefore, the expression of voltage across inductor is$v_L=-(4.56V)sin\left[\right(720rad/s\left)t\right]$