To find the reactance XLXLX_L of an inductor, imagine that a current I (t) = I0sin (ωt) I (t) = I0sin⁡ (ωt), is flowing through the inductor. What is the voltage V (t) V (t) V (t) across this inductor?

V (t) = XLI₀sin (π/2 - ωt)

Explanation:

According to Maxwell's equation which is expressed as;

V (t) = dФ/dt ... (1)

Magnetic flux Ф can also be expressed as;

Ф = LI (t)

Where

L = inductance of the inductor

I = current in Ampere

We can therefore Express Maxwell equation as:

V (t) = dLI (t) / dt ... (2)

Since the inductance is constant then voltage remains

V (t) = LdI (t) / dt

In an AC circuit, the current is time varying and it is given in the form of

I (t) = I₀sin (ωt)

Substitutes the current I (t) into equation (2)

Then the voltage across inductor will be expressed as

V (t) = Ld (I₀sin (ωt)) / dt

V (t) = LI₀ωcos (ωt)

Where cos (ωt) = sin (π/2 - ωt)

Then

V (t) = ωLI₀sin (π/2 - ωt) ... (3)

Because the voltage and current are out of phase with the phase difference of π/2 or 90°

The inductive reactance XL = ωL

Substitute ωL for XL in equation (3)

Therefore, the voltage across inductor is can be expressed as;

V (t) = XLI₀sin (π/2 - ωt)