Inductance: HRK Physics Chapter 38 Complete Guide | Self Inductance, LR Circuits & Energy Storage

Inductance: HRK Physics Chapter 38 Complete Guide | Self Inductance, LR Circuits, Energy Storage

Complete Guide to Self Inductance, LR Circuits, Energy Storage in Magnetic Fields, and Electromagnetic Oscillations

Self Inductance LR Circuits LC Oscillations Energy in Magnetic Fields Electromagnetic Induction Reading Time: 30 min

📋 Table of Contents

1. Introduction to Inductance
2. Self Inductance
3. Inductors with Magnetic Materials
4. Growth of Current in LR Circuit
- 4.1 Mathematical Derivation
- 4.2 Inductive Time Constant
5. Decay of Current in LR Circuit
6. Energy Stored in Magnetic Field
- 6.1 Derivation of Energy Formula
- 6.2 Energy Density in Magnetic Field
7. LC Circuits and Electromagnetic Oscillations
8. Growth of Current in RC Circuit
9. Practical Applications
Frequently Asked Questions

📜 Historical Background

The study of inductance dates back to the 19th century with the pioneering work of Michael Faraday and Joseph Henry. Faraday's law of electromagnetic induction (1831) established the fundamental relationship between changing magnetic fields and induced electromotive force (EMF).

Key developments include:

Michael Faraday's discovery of electromagnetic induction (1831)
Joseph Henry's independent discovery of self-inductance (1832)
Heinrich Lenz's formulation of Lenz's law (1834)
James Clerk Maxwell's unification of electricity and magnetism (1865)
Oliver Heaviside's development of operational calculus for circuit analysis (1880s)

These discoveries laid the foundation for modern electrical engineering and the development of transformers, motors, and various electronic devices.

Introduction to Inductance

🔬 What is Inductance?

Inductance is a property of an electrical conductor (typically a coil) that opposes changes in current flowing through it. When current changes, the magnetic field around the conductor also changes, inducing an electromotive force (EMF) that opposes the change in current.

This phenomenon is described by Faraday's law of electromagnetic induction and Lenz's law, which states that the induced EMF always acts to oppose the change that produced it.

📝 Types of Inductance

Self Inductance: Property of a circuit that opposes changes in its own current
Mutual Inductance: Property where changing current in one circuit induces EMF in a nearby circuit
Distributed Inductance: Inductance distributed along the length of a conductor

Self Inductance

🔬 Definition of Self Inductance

When current flows through a coil, it creates a magnetic field. If this current changes with time, the magnetic flux through the coil also changes, inducing an EMF in the coil itself. This phenomenon is called self-induction, and the property is called self-inductance.

Definition and Formula

🧮 Mathematical Formulation

Induced EMF

The induced EMF ε_L is directly proportional to the rate of change of current:

\[ \varepsilon_L \propto \frac{dI}{dt} \]

\[ \varepsilon_L = -L \frac{dI}{dt} \]

where L is the inductance of the coil.

Definition of Inductance

From the above equation, we can define inductance as:

\[ L = -\frac{\varepsilon_L}{\left(\frac{dI}{dt}\right)} \]

Unit of Inductance

The SI unit of inductance is the Henry (H):

\[ 1 \, \text{Henry} = \frac{1 \, \text{volt}}{1 \, \text{ampere/second}} = 1 \, \text{VA}^{-1}s \]

An inductor has an inductance of 1 Henry if an EMF of 1 volt is induced when the current changes at the rate of 1 ampere per second.

Inductance of a Solenoid

🧮 Derivation for Solenoid

Step 1: Magnetic Field Inside Solenoid

For a solenoid of length l with N turns carrying current I:

\[ B = \mu_0 n I \]

where n = N/l is the number of turns per unit length.

Step 2: Magnetic Flux Through One Turn

If A is the cross-sectional area:

\[ \phi_B = B \cdot A = \mu_0 n I A \]

Step 3: Total Magnetic Flux

Flux through all N turns:

\[ \phi_B = N \cdot \mu_0 n I A = (nl) \cdot \mu_0 n I A \]

\[ = \mu_0 n^2 l A I \]

Step 4: Induced EMF

According to Faraday's law:

\[ \varepsilon = -\frac{d\phi_B}{dt} = -\mu_0 n^2 l A \frac{dI}{dt} \]

Step 5: Self Inductance

Comparing with ε = -L(dI/dt):

\[ L = \mu_0 n^2 l A \]

This is the expression for self inductance of a solenoid.

Inductance of a Toroid

🧮 Derivation for Toroid

Step 1: Magnetic Field Inside Toroid

For a toroid with N turns carrying current I and mean radius r:

\[ B = \frac{\mu_0 N I}{2\pi r} \]

Step 2: Magnetic Flux Through Small Element

Considering a small area element dA = h dr:

\[ d\phi_B = B \cdot dA = B h dr \]

Step 3: Total Magnetic Flux

Integrating from inner radius a to outer radius b:

\[ \phi_B = \int_a^b N B h dr \]

\[ = \int_a^b N \left( \frac{\mu_0 N I}{2\pi r} \right) h dr \]

\[ = \frac{\mu_0 N^2 I h}{2\pi} \int_a^b \frac{dr}{r} \]

\[ = \frac{\mu_0 N^2 I h}{2\pi} \ln\left(\frac{b}{a}\right) \]

Step 4: Self Inductance

Since φ_B = LI:

\[ L = \frac{\phi_B}{I} = \frac{\mu_0 N^2 h}{2\pi} \ln\left(\frac{b}{a}\right) \]

This is the expression for self inductance of a toroid.

Inductors with Magnetic Materials

🔍 Effect of Magnetic Core

When a magnetic material (such as iron) is placed inside a solenoid, the inductance increases significantly. This is because the magnetic field inside the solenoid is enhanced by the magnetic material.

The inductance with magnetic core becomes:

L = \mu n^2 l A

where μ is the permeability of the magnetic material, which is much greater than μ₀ (permeability of free space).

💡 Key Insight: Relative Permeability

The relative permeability μ_r = μ/μ₀ indicates how much the magnetic material enhances the magnetic field compared to vacuum. For iron, μ_r can be several thousand, which means an iron-core inductor can have thousands of times more inductance than an air-core inductor of the same dimensions.

Growth of Current in LR Circuit

🔍 LR Circuit Analysis

LR Circuit Components

• Battery (E)

• Resistor (R)

• Inductor (L)

• Switch

[LR Circuit Diagram: Battery + Switch + Resistor + Inductor in series]

Circuit Behavior

When switch is closed, current doesn't reach maximum instantly
Inductor opposes the change in current
Current grows exponentially with time
Time constant τ = L/R determines growth rate

Mathematical Derivation

🧮 Current Growth Equation

Step 1: Apply Kirchhoff's Voltage Law

When switch is closed at t=0:

\[ E - IR - L\frac{dI}{dt} = 0 \]

Step 2: Rearrange the Equation

\[ L\frac{dI}{dt} + IR = E \]

\[ \frac{dI}{dt} + \frac{R}{L}I = \frac{E}{L} \]

Step 3: Solve the Differential Equation

This is a first-order linear differential equation. The solution is:

\[ I = \frac{E}{R} \left(1 - e^{-(R/L)t}\right) \]

Step 4: Final Expression

\[ I = I_0 \left(1 - e^{-t/\tau}\right) \]

where I₀ = E/R is the final steady-state current and τ = L/R is the inductive time constant.

Inductive Time Constant

📊 Understanding Time Constant

The inductive time constant τ = L/R determines how quickly the current reaches its steady-state value:

At t = τ, current reaches about 63.2% of its final value
At t = 2τ, current reaches about 86.5% of its final value
At t = 3τ, current reaches about 95.0% of its final value
At t = 5τ, current reaches about 99.3% of its final value

📈 Current Growth in LR Circuit

[Graph: Current vs Time for LR Circuit showing exponential growth]

The graph shows how current increases exponentially with time, approaching the maximum value I₀ = E/R asymptotically.

Decay of Current in LR Circuit

🧮 Current Decay Equation

Step 1: Circuit Configuration

After current reaches steady state, if we short-circuit the battery:

\[ -IR - L\frac{dI}{dt} = 0 \]

Step 2: Rearrange the Equation

\[ L\frac{dI}{dt} + IR = 0 \]

\[ \frac{dI}{dt} = -\frac{R}{L}I \]

Step 3: Solve the Differential Equation

This is an exponential decay equation. The solution is:

\[ I = I_0 e^{-(R/L)t} \]

Step 4: Final Expression

\[ I = I_0 e^{-t/\tau} \]

where I₀ is the initial current and τ = L/R is the same time constant.

📉 Current Decay in LR Circuit

[Graph: Current vs Time for LR Circuit showing exponential decay]

The graph shows how current decreases exponentially with time when the battery is removed from the circuit.

Energy Stored in Magnetic Field

🔬 Energy in Inductor

When current flows through an inductor, energy is stored in its magnetic field. This energy can be recovered when the current decreases.

Derivation of Energy Formula

🧮 Energy Stored in Inductor

Step 1: Power in Inductor

The rate at which energy is supplied to the inductor:

\[ P = \varepsilon_L I = \left(L\frac{dI}{dt}\right) I \]

Step 2: Energy Stored

The energy dU stored in time dt:

\[ dU = P dt = LI\frac{dI}{dt} dt = LI dI \]

Step 3: Total Energy

Integrating from I=0 to I=I:

\[ U = \int_0^I LI dI = \frac{1}{2}LI^2 \]

Step 4: Final Expression

\[ U = \frac{1}{2}LI^2 \]

This is the energy stored in the magnetic field of an inductor carrying current I.

Energy Density in Magnetic Field

🧮 Energy Density Derivation

Step 1: For a Solenoid

We know that for a solenoid:

\[ L = \mu_0 n^2 l A \]

\[ B = \mu_0 n I \]

Step 2: Express Energy in Terms of B

\[ U = \frac{1}{2}LI^2 = \frac{1}{2}(\mu_0 n^2 l A)\left(\frac{B}{\mu_0 n}\right)^2 \]

\[ = \frac{1}{2}\mu_0 n^2 l A \cdot \frac{B^2}{\mu_0^2 n^2} \]

\[ = \frac{1}{2}\frac{B^2}{\mu_0} l A \]

Step 3: Energy Density

Energy per unit volume:

\[ u_B = \frac{U}{\text{Volume}} = \frac{U}{lA} = \frac{1}{2}\frac{B^2}{\mu_0} \]

Step 4: General Form

In general, for any magnetic field:

\[ u_B = \frac{1}{2}\frac{B^2}{\mu_0} \]

This is the energy density in a magnetic field.

💡 Comparison with Electric Field Energy

Notice the similarity between energy densities in electric and magnetic fields:

u_E = \frac{1}{2}\varepsilon_0 E^2

u_B = \frac{1}{2}\frac{B^2}{\mu_0}

This symmetry reflects the deep connection between electricity and magnetism in Maxwell's equations.

LC Circuits and Electromagnetic Oscillations

🔍 LC Circuit Analysis

LC Circuit Components

• Capacitor (C)

• Inductor (L)

• Switch

[LC Circuit Diagram: Capacitor + Inductor in series with switch]

Circuit Behavior

Energy oscillates between capacitor and inductor
Current and charge vary sinusoidally with time
Natural frequency depends on L and C
No energy loss in ideal LC circuit

Qualitative Analysis

Step 1: Initial Condition

Capacitor is fully charged with charge Q₀. All energy is stored in the electric field of the capacitor.

Step 2: Discharge Through Inductor

Capacitor discharges through inductor. Current builds up, storing energy in the magnetic field of the inductor.

Step 3: Current Continues to Flow

Even when capacitor is fully discharged, current continues due to inductor's opposition to change. This charges capacitor in opposite direction.

Step 4: Reverse Process

The process repeats in reverse direction, creating continuous oscillations.

Quantitative Analysis

🧮 LC Circuit Equations

Step 1: Apply Kirchhoff's Voltage Law

\[ \frac{q}{C} - L\frac{dI}{dt} = 0 \]

Step 2: Express in Terms of Charge

Since I = dq/dt, we have:

\[ \frac{q}{C} + L\frac{d^2q}{dt^2} = 0 \]

\[ \frac{d^2q}{dt^2} + \frac{1}{LC}q = 0 \]

Step 3: Identify Equation Type

This is the simple harmonic motion equation with:

\[ \omega^2 = \frac{1}{LC} \]

Step 4: Solution

The general solution is:

\[ q = Q_0 \cos(\omega t + \phi) \]

where Q₀ is the maximum charge and φ is the phase constant.

Step 5: Current Expression

\[ I = \frac{dq}{dt} = -\omega Q_0 \sin(\omega t + \phi) \]

\[ = -I_0 \sin(\omega t + \phi) \]

where I₀ = ωQ₀ is the maximum current.

📊 Important Parameters

Angular Frequency: ω = 1/√(LC)
Frequency: f = ω/(2π) = 1/(2π√(LC))
Period: T = 1/f = 2π√(LC)
Total Energy: U = ½(Q₀²/C) = ½LI₀² (constant)

Damped and Forced Oscillations

🔬 Real LC Circuits

In real circuits, resistance causes damping of oscillations:

Damped Oscillations: Amplitude decreases exponentially with time due to energy loss in resistance
Forced Oscillations: External AC source can maintain oscillations at driving frequency
Resonance: Maximum amplitude occurs when driving frequency equals natural frequency

Growth of Current in RC Circuit

🧮 Comparison with RC Circuit

Step 1: RC Circuit Equation

For an RC circuit with battery E, resistor R, and capacitor C:

\[ E - IR - \frac{q}{C} = 0 \]

Step 2: Express in Terms of Charge

Since I = dq/dt:

\[ E - R\frac{dq}{dt} - \frac{q}{C} = 0 \]

\[ R\frac{dq}{dt} + \frac{q}{C} = E \]

Step 3: Solution

The solution for charge as a function of time is:

\[ q = CE\left(1 - e^{-t/RC}\right) \]

Step 4: Current

\[ I = \frac{dq}{dt} = \frac{E}{R}e^{-t/RC} \]

Note that in RC circuit, current decreases exponentially, while in LR circuit during growth, current increases exponentially.

Practical Applications

🔌 Transformers

Inductance is fundamental to transformer operation, which is essential for power transmission and voltage conversion in electrical systems.

📡 Radio Tuners

LC circuits are used as tuners in radios and televisions to select specific frequencies from the electromagnetic spectrum.

⚡ Inductive Sensors

Inductance principles are used in proximity sensors, metal detectors, and various industrial automation applications.

🔋 Power Supplies

Inductors are used in switching power supplies to smooth current and reduce ripple in DC power conversion.

🎛️ Filters

LC circuits form the basis of various electronic filters used in signal processing to remove unwanted frequency components.

⚙️ Electric Motors

Inductance plays a crucial role in the operation of electric motors, particularly in the windings that create magnetic fields.

Frequently Asked Questions

❓ Why does an inductor oppose changes in current but not steady current?

An inductor generates a back EMF only when the current through it is changing. According to Faraday's law, a changing current produces a changing magnetic field, which induces an EMF that opposes the change (Lenz's law). When current is steady, the magnetic field is constant, so no EMF is induced, and the inductor behaves like an ordinary conductor.

❓ What happens to the energy stored in an inductor when the circuit is opened?

When a circuit with significant inductance is opened suddenly, the rapid change in current induces a large voltage spike. This can cause sparking at the switch contacts. In practical circuits, diodes or other protective devices are often used to provide a path for the inductor current to decay gradually, safely dissipating the stored energy.

❓ How does the presence of a magnetic core affect inductance?

Aspect	Air Core	Magnetic Core
Inductance	Lower	Higher (μ_r times)
Saturation	No saturation	Can saturate at high currents
Energy Loss	Lower (mainly resistive)	Higher (hysteresis & eddy currents)
Frequency Response	Good for high frequencies	Limited at high frequencies

📚 Master Electromagnetism

Understanding inductance is crucial for advanced studies in electrical engineering, electronics, and electromagnetic theory. Continue your journey into the fascinating world of electromagnetism.

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

Author: Muhammad Ali Malik | Physics Educator | Contact: +923016775811