Talha's Physics Academy
Flow of AC Through an Inductor
Video Lecture
Watch the complete video lecture below to visually understand how alternating current flows through an inductor, back EMF, phase relationships, and inductive reactance.
Describe the Flow of AC Through an Inductor
Consider an inductor in the form of a solenoid connected to an AC source. When the circuit is closed, alternating current starts to flow through the inductor. Since the magnitude and direction of the current change continuously, the associated magnetic field also varies. This varying magnetic field sets up an induced electromotive force (EMF)—commonly known as back EMF—in the inductor, which opposes the change in accordance with Lenz's law. The magnitude of this induced EMF is given by:
$$\varepsilon = L \frac{di}{dt}$$
Therefore, to sustain the current, the applied voltage must match the back EMF. Thus, the magnitude of the voltage supplied to the coil is expressed as:
$$v = L \frac{di}{dt} \quad \text{--- (i)}$$
The alternating voltage produces a sinusoidal current given by:
$$i = i_0 \sin(\omega t)$$
Substituting the current equation into equation (i) yields:
$$v = L \frac{d}{dt} [i_0 \sin(\omega t)]$$
$$v = L i_0 \omega \cos(\omega t)$$
Using trigonometric identities ($\cos(\theta) = \sin(\theta + \frac{\pi}{2})$), we can rewrite this as:
$$v = L i_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)$$
Rearranging the terms:
$$v = i_0 (L\omega) \sin\left(\omega t + \frac{\pi}{2}\right)$$
Where inductive reactance is defined as $X_L = L\omega$, so substituting $X_L$ into the equation gives:
$$v = i_0 X_L \sin\left(\omega t + \frac{\pi}{2}\right)$$
$$v = v_0 \sin\left(\omega t + \frac{\pi}{2}\right)$$
Phase Relationship
This equation shows that in a pure inductive circuit, the voltage exhibits sinusoidal variation and it leads the current by $90^\circ$ (or the current lags the voltage by $90^\circ$).
The reason for voltage leading in an inductive circuit is the generation of induced electromotive force (back EMF) when an alternating voltage is applied. This back EMF appears instantaneously and induces a counter-current, introducing a brief time delay (typically in the order of milliseconds). Consequently, there is a time lag for the circuit current to overcome this opposing current and attain its maximum value. That is why voltage manifests first, followed by the appearance of the current after a short interval.
Reactance of Inductor ($X_L$)
In a purely inductive circuit, inductive reactance represents the opposition to alternating current flow. Similar to resistance, reactance is measured in Ohms ($\Omega$), but it is denoted by the symbol $X_L$ to differentiate it from purely resistive values:
$$X_L = \omega L = 2\pi f L$$
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