To understand the concept of reactance (of an inductor) and its frequency dependence.

When an inductor is connected to a voltage source that varies sinusoidally, a sinusoidal current will flow through the inductor, its magnitude depending on the frequency. This is the essence of AC (alternating current) circuits used in radio, TV, and stereos. Circuit elements like inductors, capacitors, and resistors are linear devices, so the amplitude I0 of the current will be proportional to the amplitude V0 of the voltage. However, the current and voltage may not be in phase with each other. This new relationship between voltage and current is summarized by the reactance, the ratio of voltage and current amplitudes, V0, and I0: XL=V0/I0, where the subscript L indicates that this formula applies to an inductor.

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

Express your answer in terms of I0, ?, and the inductance L.

The voltage V(t) across the inductor : V(t) = I₀ ωLsinωt

Given that :

Voltage across inductor leads current by 90° in an R-L-C  circuit ( AC )

v(t) = v₀sinωt   ( 1 )

where :  ω = angular velocity

Note: XL ( inductive reactance ) = v₀ / I₀    

∴ v₀ = XL * I₀   ( 2 )  

where;  XL = ωL

Back to equation ( 1 )

V(t) =  I₀ ωLsinωt

Hence we can conclude that the voltage V(t) across the inductor : V(t) = I₀ ωLsinωt

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The voltage across the inductor I'm terms of I0, ω and L is expressed as V(t) = I0ωLsinωt

Explanation:

In an R-L AC circuit, the voltage across the inductor leads the current by 90°.

Voltage V(t) across the inductor can be expressed as

V(t) = V0sinωt ... (1) where;

ω is the angular velocity which is a function of its frequency.

Since the inductive reactance

XL = V0/I0

V0 = I0XL

also if XL = ωL

V0 = I0(ωL)...(2)

Substituting equation 2 into 1 to get V(t),

V(t) = I0ωLsin(ωt)

V(t) = I0ωLsinωt

The voltage across the inductor in terms of I0, ω and L is expressed as V(t) = I0ωLsinωt

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