# A.C Through an Inductor An *inductor* is an electrical component, typically a coil of wire, that opposes changes in the electric current passing through it. This section explores its behavior in an Alternating Current (A.C.) circuit. ## Inductor in an A.C. Circuit When an alternating voltage is applied across an ideal inductor (with zero resistance), the changing current induces a *back e.m.f.* ($\varepsilon$) that opposes the change. The applied voltage must be equal in magnitude to this back e.m.f. at every instant. - Applied Voltage equals Back e.m.f. - Back e.m.f. formula: $$ \varepsilon = L \frac{\Delta I}{\Delta t} $$ where $L$ is the inductance. ## Voltage and Current Phase Relationship 1. Let the alternating current through the inductor be represented by: $$ I = I_0 \sin \omega t $$ 2. The applied voltage $V$ required to overcome the back e.m.f. is: $$ V = L \frac{\Delta I}{\Delta t} = L \frac{\Delta(I_0 \sin \omega t)}{\Delta t} $$ 3. Solving the derivative, we get: $$ V = \omega L I_0 \cos \omega t $$ 4. This can be expressed in terms of sine to compare with the current equation, using the identity $\cos \theta = \sin(\theta + \frac{\pi}{2})$: $$ V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right) $$ where the peak voltage is $V_0 = \omega L I_0$. By comparing the equations for current and voltage, we conclude that in a purely inductive circuit, the **voltage leads the current by $\frac{\pi}{2}$ radians or $90^{\circ}$**. - Waveform Diagram: ## Inductive Reactance ($X_L$) - **Definition:** The opposition offered by an inductor to the flow of A.C. is called inductive reactance. It is analogous to resistance in Ohm's law. - **Formula:** From the peak voltage equation $V_0 = I_0 (\omega L)$, we can define reactance $X_L$ as: $$ X_L = \frac{V_0}{I_0} = \omega L $$ or in terms of frequency $f$ ($\omega = 2\pi f$): $$ X_L = 2\pi f L $$ - **Properties:** - $X_L$ is directly proportional to the frequency ($f$) of the A.C. - For D.C. circuits, $f = 0$, so $X_L = 0$. An ideal inductor has zero opposition to a steady direct current. - The unit for inductive reactance is the **Ohm ($\Omega$)**. ## Power Dissipation In a purely inductive circuit, the average power dissipated over a complete cycle is zero. This is because the energy is stored in the magnetic field during one half-cycle and returned to the source during the next half-cycle. | Formula | Description | | :--- | :--- | | $\varepsilon = L \frac{\Delta I}{\Delta t}$ | Back e.m.f. in an inductor. | | $V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$ | Voltage across an inductor (when $I = I_0 \sin \omega t$). | | $X_L = \omega L = 2\pi f L$ | Inductive reactance. | | $V_0 = I_0 X_L$ | Relationship between peak voltage, current, and reactance. | Inductors are crucial components in circuits like filters, oscillators, and transformers, where controlling current based on frequency is essential.