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AC Voltage Applied to an Inductor

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Estimated time: 8 minutes
CBSE: Class 12

Introduction

When an alternating voltage is connected across a pure inductor, the inductor continuously opposes changes in current due to self-induction. Unlike a resistor, a pure (ideal) inductor has zero resistance — it only has inductance L.

Maharashtra State Board: Class 11

Definition: Inductive Reactance

The effective resistance offered by an inductor to the alternating current is called inductive reactance.

Maharashtra State Board: Class 11

Formula: Inductive Reactance

XL ​= 2πfL (∝ f)

CBSE: Class 12

Derivation

Given: v = vm sin ⁡ωt applied across a pure inductor of inductance L.

Apply Kirchhoff's Voltage Law:

  • \[v-L\frac{di}{dt}=0\] ⇒ L\[\frac {di}{dt}\] = vm sin ωt

Separate and integrate:

  • \[\frac{di}{dt}=\frac{v_m}{L}\sin\omega t\]
  • i = \[\int\frac{v_m}{L}\sin\omega tdt=-\frac{v_m}{\omega L}\cos\omega t+C\]

Taking integration constant C = 0 (no DC offset in a purely AC circuit):

  • i = -\[\frac {v_m}{ωL}\] cos ωt

Convert using trigonometric identity:

Using -cos θ = sin (θ - \[\frac {\pi}{2}\])

  • i = \[{i_{m}\sin\left(\omega t-\frac{\pi}{2}\right)}\]

where

  • \[{i_m=\frac{v_m}{\omega L}}\]

Define Inductive Reactance XL​:

  • XL ​= ωL = 2πfL​

Therefore: \[i_m=\frac{v_m}{X_L}\]

This mirrors Ohm's Law (I = V/R), with XL​ playing the role of resistance.

CBSE: Class 12

Example

Problem: A pure inductor of 25.0 mH is connected to a source of 220 V, 50 Hz. Find the inductive reactance and rms current.

Given:

  • L = 25.0 mH = 25.0 × 10−3 H
  • Vrms = 220 V
  • f = 50 Hz

Formula:

XL = 2πfL
Irms = \[\frac {V_{rms}}{X_L}\]

Solution:

XL = 2π × 50 × 25.0 × 10−3
XL = 2 × 3.14 × 50 × 0.025 = 7.85 Ω
Irms = \[\frac {220}{7.85}\] ≈ 28.0 A

Answer: Inductive Reactance XL ≈ 7.85 Ω; RMS Current Irms ≈ 28.0 A

CBSE: Class 12

Real-World Applications

Application How This Concept Is Used
Transformers Transformer cores operate based on the principle of pure inductance, enabling efficient transfer of electrical energy between coils.
Inductive Filters Inductors block high-frequency noise while allowing low-frequency or DC components to pass, making them useful in power supply filters.
Wireless Charging Electrical energy is transferred wirelessly between coils through the principle of mutual inductance.

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