Name: Displacement Current
Uploaded: 2022-12-19T08:17:13-05:00
Description: Ampère's law, in its usual form, does not work in places where the current changes with time and is not steady.

Chapter 30: Electromagnetic Induction Back To Chapter

30.6: Displacement Current

TABLE OF
CONTENTS

X

Chapter 1: Units, Dimensions, and Measurements

30
1.1: The Scope of Physics
30
1.2: Orders of Magnitude
30
1.3: Units and Standards of Measurement
30
1.4: Base Quantities and Derived Quantities
30
1.5: Conversion of Units
30
1.6: Accuracy and Precision
30
1.7: Random and Systematic Errors
30
1.8: Rules for Significant Figures
30
1.9: Significant Figures in Calculations
30
1.10: Dimensional Analysis
30
1.11: Solving Problems in Physics

Chapter 2: Vectors and Scalars

30
2.1: Introduction to Scalars
30
2.2: Introduction to Vectors
30
2.3: Vector Components in the Cartesian Coordinate System
30
2.4: Polar and Cylindrical Coordinates
30
2.5: Vector Algebra: Graphical Method
30
2.6: Vector Algebra: Method of Components
30
2.7: Scalar Product (Dot Product)
30
2.8: Vector Product (Cross Product)

Chapter 3: Motion Along a Straight Line

30
3.1: Position and Displacement
30
3.2: Average Velocity
30
3.3: Instantaneous Velocity - I
30
3.4: Instantaneous Velocity - II
30
3.5: Average Acceleration
30
3.6: Instantaneous Acceleration
30
3.7: Kinematic Equations - I
30
3.8: Kinematic Equations - II
30
3.9: Kinematic Equations - III
30
3.10: Kinematic Equations: Problem Solving
30
3.11: Free-falling Bodies: Introduction
30
3.12: Free-falling Bodies: Example
30
3.13: Velocity and Position by Graphical Method
30
3.14: Velocity and Position by Integral Method

Chapter 4: Motion in Two or Three Dimensions

30
4.1: Position and Displacement Vectors
30
4.2: Average and Instantaneous Velocity Vectors
30
4.3: Acceleration Vectors
30
4.4: Direction of Acceleration Vectors
30
4.5: Projectile Motion
30
4.6: Projectile Motion: Equations
30
4.7: Projectile Motion: Example
30
4.8: Uniform Circular Motion
30
4.9: Non-uniform Circular Motion
30
4.10: Relative Velocity in One Dimension
30
4.11: Relative Velocity in Two Dimensions

Chapter 5: Newton's Laws of Motion

30
5.1: Force
30
5.2: Types of Forces
30
5.3: Newton's First Law: Introduction
30
5.4: Newton's First Law: Application
30
5.5: Internal and External Forces
30
5.6: Newton's Second Law
30
5.7: Mass and Weight
30
5.8: Weightlessness
30
5.9: Newton's Third Law: Introduction
30
5.10: Newton's Third Law: Examples
30
5.11: Drawing Free-body Diagrams: Rules
30
5.12: Free Body Diagrams: Examples
30
5.13: Inertial Frames of Reference
30
5.14: Non-inertial Frames of Reference

Chapter 6: Application of Newton's Laws of Motion

30
6.1: First Law: Particles in One-dimensional Equilibrium
30
6.2: First Law: Particles in Two-dimensional Equilibrium
30
6.3: Second Law: Motion under Same Force
30
6.4: Second Law: Motion under Same Acceleration
30
6.5: Frictional Force
30
6.6: Static and Kinetic Frictional Force
30
6.7: Dynamics of Circular Motion
30
6.8: Dynamics Of Circular Motion: Applications

Chapter 7: Work and Kinetic Energy

30
7.1: Work
30
7.2: Positive, Negative, and Zero Work
30
7.3: Energy
30
7.4: Types of Potential Energy
30
7.5: Types of Kinetic Energy
30
7.6: Kinetic Energy - I
30
7.7: Kinetic Energy - II
30
7.8: Work-energy Theorem
30
7.9: Power
30
7.10: Power Expended by a Constant Force

Chapter 8: Potential Energy and Energy Conservation

30
8.1: Gravitational Potential Energy
30
8.2: Elastic Potential Energy
30
8.3: Conservative Forces
30
8.4: Non-conservative Forces
30
8.5: Conservation of Energy
30
8.6: Conservation of Energy: Application
30
8.7: Force and Potential Energy in One Dimension
30
8.8: Force and Potential Energy in Three Dimensions
30
8.9: Energy Diagrams - I
30
8.10: Energy Diagrams - II

Chapter 9: Linear Momentum, Impulse and Collisions

30
9.1: Linear Momentum
30
9.2: Force and Momentum
30
9.3: Impulse
30
9.4: Impulse-Momentum Theorem
30
9.5: Conservation of Momentum: Introduction
30
9.6: Conservation of Momentum: Problem Solving
30
9.7: Types Of Collisions - I
30
9.8: Types of Collisions - II
30
9.9: Elastic Collisions: Introduction
30
9.10: Elastic Collisions: Case Study
30
9.11: Collisions in Multiple Dimensions: Introduction
30
9.12: Collisions in Multiple Dimensions: Problem Solving
30
9.13: Center of Mass: Introduction
30
9.14: Significance of Center of Mass
30
9.15: Rocket Propulsion in Empty Space - I
30
9.16: Rocket Propulsion In Empty Space - II
30
9.17: Rocket Propulsion in Gravitational Field - I
30
9.18: Rocket Propulsion in Gravitational Field - II

Chapter 10: Rotation and Rigid Bodies

30
10.1: Angular Velocity and Displacement
30
10.2: Angular Velocity and Acceleration
30
10.3: Rotation with Constant Angular Acceleration - I
30
10.4: Rotation with Constant Angular Acceleration - II
30
10.5: Relating Angular And Linear Quantities - I
30
10.6: Relating Angular And Linear Quantities - II
30
10.7: Moment of Inertia
30
10.8: Moment of Inertia and Rotational Kinetic Energy
30
10.9: Moment of Inertia: Calculations
30
10.10: Parallel-axis Theorem
30
10.11: Moment of Inertia of Compound Objects

Chapter 11: Dynamics of Rotational Motions

30
11.1: Torque
30
11.2: Net Torque Calculations
30
11.3: Work and Power for Rotational Motion
30
11.4: Work-Energy Theorem for Rotational Motion
30
11.5: Angular Momentum: Single Particle
30
11.6: Angular Momentum: Rigid Body
30
11.7: Conservation of Angular Momentum
30
11.8: Conservation of Angular Momentum: Application
30
11.9: Gyroscope
30
11.10: Gyroscope: Precession

Chapter 12: Equilibrium and Elasticity

30
12.1: Static Equilibrium - I
30
12.2: Static Equilibrium - II
30
12.3: Center of Gravity
30
12.4: Finding the Center of Gravity
30
12.5: Rigid Body Equilibrium Problems - I
30
12.6: Rigid Body Equilibrium Problems - II
30
12.7: Stress
30
12.8: Strain and Elastic Modulus
30
12.9: Elasticity
30
12.10: Plasticity

Chapter 13: Fluid Mechanics

30
13.1: Density
30
13.2: Pressure of Fluids
30
13.3: Pascal's Law
30
13.4: Application of Pascal's Law
30
13.5: Buoyancy
30
13.6: Archimedes' Principle
30
13.7: Density and Archimedes' Principle
30
13.8: Laminar and Turbulent Flow
30
13.9: Equation of Continuity
30
13.10: Bernoulli's Equation
30
13.11: Bernoulli's Principle
30
13.12: Viscosity
30
13.13: Poiseuille's Law and Reynolds Number

Chapter 14: Gravitation

30
14.1: Newton's Law of Gravitation
30
14.2: Gravity between Spherical Bodies
30
14.3: Acceleration due to Gravity on Earth
30
14.4: Acceleration due to Gravity on Other Planets
30
14.5: Apparent Weight and the Earth's Rotation
30
14.6: Variation in Acceleration due to Gravity near the Earth's Surface
30
14.7: Potential Energy due to Gravitation
30
14.8: Escape Velocity
30
14.9: Circular Orbits and Critical Velocity for Satellites
30
14.10: Energy of a Satellite in a Circular Orbit
30
14.11: Kepler's First Law of Planetary Motion
30
14.12: Kepler's Second Law of Planetary Motion
30
14.13: Kepler's Third Law of Planetary Motion
30
14.14: Tidal Forces
30
14.15: Schwarzschild Radius and Event Horizon
30
14.16: Detection of Black Holes
30
14.17: Principle of Equivalence
30
14.18: Space-Time Curvature and the General Theory of Relativity

Chapter 15: Oscillations

30
15.1: Simple Harmonic Motion
30
15.2: Characteristics of Simple Harmonic Motion
30
15.3: Oscillations about an Equilibrium Position
30
15.4: Energy in Simple Harmonic Motion
30
15.5: Simple Harmonic Motion and Uniform Circular Motion
30
15.6: Simple Pendulum
30
15.7: Torsional Pendulum
30
15.8: Damped Oscillations
30
15.9: Types of Damping
30
15.10: Forced Oscillations
30
15.11: Concept of Resonance and its Characteristics

Chapter 16: Waves

30
16.1: Travelling Waves
30
16.2: Wave Parameters
30
16.3: Equations of Wave Motion
30
16.4: Velocity and Acceleration of a Wave
30
16.5: Kinetic and Potential Energy of a Wave
30
16.6: Energy and Power of a Wave
30
16.7: Interference and Superposition of Waves
30
16.8: Reflection of Waves
30
16.9: Propagation of Waves
30
16.10: Standing Waves
30
16.11: Modes of Standing Waves - I

Chapter 17: Sound

30
17.1: Sound Waves
30
17.2: Perception of Sound Waves
30
17.3: Speed of Sound in Solids and Liquids
30
17.4: Speed of Sound in Gases
30
17.5: Sound Intensity
30
17.6: Sound Intensity Level
30
17.7: Sound Waves: Interference
30
17.8: Sound Waves: Resonance
30
17.9: Doppler Effect - I
30
17.10: Doppler Effect - II
30
17.11: Shock Waves

Chapter 18: Temperature and Heat

30
18.1: Temperature and Thermal Equilibrium
30
18.2: Zeroth Law of Thermodynamics
30
18.3: Thermometers and Temperature Scales
30
18.4: Gas Thermometers and the Kelvin Scale
30
18.5: Thermal Expansion
30
18.6: Thermal Stress
30
18.7: Heat Flow and Specific Heat
30
18.8: Phase Changes
30
18.9: Mechanisms of Heat Transfer I
30
18.10: Mechanisms of Heat Transfer II

Chapter 19: The Kinetic Theory of Gases

30
19.1: Ideal Gas Equation
30
19.2: Van der Waals Equation
30
19.3: pV-Diagrams
30
19.4: Kinetic Theory of an Ideal Gas
30
19.5: Molecular Kinetic Energy
30
19.6: Distribution of Molecular Speeds
30
19.7: Phase Diagram

Chapter 20: The First Law of Thermodynamics

30
20.1: Thermodynamic Systems
30
20.2: Work Done During Volume Change
30
20.3: Path Between Thermodynamics States
30
20.4: Internal Energy
30
20.5: First Law of Thermodynamics
30
20.6: Cyclic Processes And Isolated Systems
30
20.7: Isothermal Processes
30
20.8: Isochoric and Isobaric Processes
30
20.9: Heat Capacities of an Ideal Gas I
30
20.10: Heat Capacities of an Ideal Gas II
30
20.11: Heat Capacities of an Ideal Gas III
30
20.12: Adiabatic Processes for an Ideal Gas
30
20.13: Pressure and Volume in an Adiabatic Process
30
20.14: Work Done in an Adiabatic Process

Chapter 21: The Second Law of Thermodynamics

30
21.1: Reversible and Irreversible Processes
30
21.2: Heat Engines
30
21.3: Refrigerators and Heat Pumps
30
21.4: Statements of the Second Law of Thermodynamics
30
21.5: The Carnot Cycle
30
21.6: Efficiency of The Carnot Cycle
30
21.7: The Carnot Cycle and the Second Law of Thermodynamics
30
21.8: Entropy
30
21.9: Entropy Change in Reversible Processes
30
21.10: Entropy and the Second Law of Thermodynamics

Chapter 22: Electric Charges and Fields

30
22.1: Electric Charges
30
22.2: Sources and Properties of Electric Charge
30
22.3: Conductors and Insulators
30
22.4: Charging Conductors By Induction
30
22.5: Coulomb's Law
30
22.6: Coulomb's Law and The Principle of Superposition
30
22.7: Electric Field
30
22.8: Electric Field of Two Equal and Opposite Charges
30
22.9: Continuous Charge Distributions
30
22.10: Electric Field Lines
30
22.11: Properties of Electric Field Lines
30
22.12: Electric Dipoles and Dipole Moment
30
22.13: Induced Electric Dipoles

Chapter 23: Gauss's Law

30
23.1: Electric Flux
30
23.2: Gauss's Law
30
23.3: Gauss's Law: Spherical Symmetry
30
23.4: Gauss's Law: Cylindrical Symmetry
30
23.5: Gauss's Law: Planar Symmetry
30
23.6: Electric Field Inside a Conductor
30
23.7: Charge on a Conductor
30
23.8: Electric Field at the Surface of a Conductor

Chapter 24: Electric Potential

30
24.1: Electric Potential Energy
30
24.2: Electric Potential Energy in a Uniform Electric Field
30
24.3: Electric Potential Energy of Two Point Charges
30
24.4: Electric Potential and Potential Difference
30
24.5: Finding Electric Potential From Electric Field
30
24.6: Equipotential Surfaces and Field Lines
30
24.7: Equipotential Surfaces and Conductors
30
24.8: Determining Electric Field From Electric Potential

Chapter 25: Capacitance

30
25.1: Capacitors and Capacitance
30
25.2: Spherical and Cylindrical Capacitor
30
25.3: Capacitors in Series and Parallel
30
25.4: Energy Stored in a Capacitor
30
25.5: Capacitor With A Dielectric
30
25.6: Dielectric Polarization in a Capacitor

Chapter 26: Current and Resistance

30
26.1: Electrical Current
30
26.2: Drift Velocity
30
26.3: Current Density
30
26.4: Resistivity
30
26.5: Resistance
30
26.6: Ohm's Law
30
26.7: Electrical Power

Chapter 27: Direct-Current Circuits

30
27.1: Electromotive Force
30
27.2: Resistors In Series
30
27.3: Resistors In Parallel
30
27.4: Kirchhoff's Rules
30
27.5: Galvanometer
30
27.6: Ammeter
30
27.7: RC Circuits: Charging A Capacitor
30
27.8: RC Circuits: Discharging A Capacitor

Chapter 28: Magnetic Forces and Fields

30
28.1: Magnetism
30
28.2: Magnetic Fields
30
28.3: Magnetic Field Lines
30
28.4: Magnetic Flux
30
28.5: Motion Of A Charged Particle In A Magnetic Field
30
28.6: Magnetic Force On A Current-Carrying Conductor
30
28.7: Force On A Current Loop In A Magnetic Field
30
28.8: Torque On A Current Loop In A Magnetic Field

Chapter 29: Sources of Magnetic Fields

30
29.1: Magnetic Field due to Moving Charges
30
29.2: Biot-Savart Law
30
29.3: Magnetic Field Due To A Thin Straight Wire
30
29.4: Magnetic Force Between Two Parallel Currents
30
29.5: Magnetic Field Of A Current Loop
30
29.6: Ampere's Law
30
29.7: Ampere's Law: Problem-Solving
30
29.8: Solenoids
30
29.9: Magnetic Field of a Solenoid
30
29.10: Toroids
30
29.11: Diamagnetism
30
29.12: Paramagnetism
30
29.13: Ferromagnetism

Chapter 30: Electromagnetic Induction

30
30.1: Induction
30
30.2: Faraday's Law
30
30.3: Lenz's Law
30
30.4: Motional Emf
30
30.5: Induced Electric Fields
30
30.6: Displacement Current
30
30.7: Significance of Displacement Current
30
30.8: Electromagnetic Fields
30
30.9: Maxwell's Equation Of Electromagnetism
30
30.10: Symmetry in Maxwell's Equations
30
30.11: Electric Generator: Alternator
30
30.12: Back EMF

Chapter 31: Inductance

30
31.1: Mutual Inductance
30
31.2: Self-Inductance
30
31.3: Inductors
30
31.4: Energy In A Magnetic Field
30
31.5: RL Circuits
30
31.6: Current Growth And Decay In RL Circuits
30
31.7: LC Circuits
30
31.8: Oscillations In An LC Circuit
30
31.9: RLC Series Circuits

Chapter 32: Alternating-Current Circuits

30
32.1: AC Sources
30
32.2: Resistor in an AC Circuit
30
32.3: Capacitor in an AC Circuit
30
32.4: Inductor in an AC Circuit
30
32.5: RLC Series Circuits: Introduction
30
32.6: RLC Series Circuits: Impedance
30
32.7: Power in an AC Circuit
30
32.8: Resonance in an AC Circuit

Chapter 33: Electromagnetic Waves

30
33.1: Electromagnetic Waves
30
33.2: The Electromagnetic Spectrum
30
33.3: Plane Electromagnetic Waves I
30
33.4: Plane Electromagnetic Waves II
30
33.5: Propagation Speed of Electromagnetic Waves
30
33.6: Electromagnetic Waves in Matter
30
33.7: Energy Carried By Electromagnetic Waves
30
33.8: Intensity Of Electromagnetic Waves
30
33.9: Momentum And Radiation Pressure

Full Table of Contents

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TRANSCRIPT

Ampere's law states that the line integral of a magnetic field along a closed curve equals the permeability of free space times the net current passing through the loop.

Consider a parallel plate capacitor connected with a battery in a charging condition.

Applying Ampere's law to the shown Amperian loop with two surfaces gives two different magnetic field values, which is impossible.

As the capacitor is charging, the electric field, and hence the electric flux, increases through the bulging surface. The value of electric flux can be obtained in terms of the charge.

In 1865, James Clerk Maxwell, predicted that due to this time-varying electric field, a non-zero magnetic field is produced between the plates of the capacitor through a fictitious current called displacement current.

The expression for displacement current is given in terms of electric flux.

Thus, Ampere's law is modified with the inclusion of an additional term for displacement current. It is known as the Ampere-Maxwell law or generalized Ampere's law.

30.6: Displacement Current

Ampère's law, in its usual form, does not work in places where the current changes with time and is not steady. Thus, Maxwell suggested including an additional contribution, called the displacement current, I_d, to the real conduction current I.

Equation1

where the displacement current is defined to be

Equation2

Here, ε₀is the permittivity of free space, and Φ_E is the electric flux.

The displacement current is an extra term in Maxwell's equations that is analogous to a real current in Ampère's law. However, it is produced by a changing electric field. It accounts for a changing electric field producing a magnetic field, just as a real current does, but the displacement current can produce a magnetic field even when no real current is present. When this extra term is included, the modified Ampère's law equation becomes

Equation3

In this way, Ampère's law can be modified so that it works in all situations, and it is independent of the surface through which the current I is measured.

Suggested Reading

Chapter 1: Units, Dimensions, and Measurements

Chapter 2: Vectors and Scalars

Chapter 3: Motion Along a Straight Line

Chapter 4: Motion in Two or Three Dimensions

Chapter 5: Newton's Laws of Motion

Chapter 6: Application of Newton's Laws of Motion

Chapter 7: Work and Kinetic Energy

Chapter 8: Potential Energy and Energy Conservation

Chapter 9: Linear Momentum, Impulse and Collisions

Chapter 10: Rotation and Rigid Bodies

Chapter 11: Dynamics of Rotational Motions

Chapter 12: Equilibrium and Elasticity

Chapter 13: Fluid Mechanics

Chapter 14: Gravitation

Chapter 15: Oscillations

Chapter 16: Waves

Chapter 17: Sound

Chapter 18: Temperature and Heat

Chapter 19: The Kinetic Theory of Gases

Chapter 20: The First Law of Thermodynamics

Chapter 21: The Second Law of Thermodynamics

Chapter 22: Electric Charges and Fields

Chapter 23: Gauss's Law

Chapter 24: Electric Potential

Chapter 25: Capacitance

Chapter 26: Current and Resistance

Chapter 27: Direct-Current Circuits

Chapter 28: Magnetic Forces and Fields

Chapter 29: Sources of Magnetic Fields

Chapter 30: Electromagnetic Induction

Chapter 31: Inductance

Chapter 32: Alternating-Current Circuits

Chapter 33: Electromagnetic Waves

30.6: Displacement Current

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