Name: Ampere-Maxwell's Law: Problem-Solving
Uploaded: 2023-09-07T11:48:17-04:00
Duration: 1 min 17 s
Description: A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V.

Chapter 30: Electromagnetic Induction Back To Chapter

30.14: Ampere-Maxwell's Law: Problem-Solving

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: Models, Theories, and Laws
30
1.4: Units and Standards of Measurement
30
1.5: Estimation of the Physical Quantities
30
1.6: Base Quantities and Derived Quantities
30
1.7: Conversion of Units
30
1.8: Accuracy and Precision
30
1.9: Random and Systematic Errors
30
1.10: Rules for Significant Figures
30
1.11: Significant Figures in Calculations
30
1.12: Dimensional Analysis
30
1.13: Problem Solving: Dimensional Analysis
30
1.14: 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: Spherical Coordinates
30
2.6: Vector Algebra: Graphical Method
30
2.7: Vector Algebra: Method of Components
30
2.8: Scalar Product (Dot Product)
30
2.9: Vector Product (Cross Product)
30
2.10: Scalar and Vector Triple Products
30
2.11: Gradient and Del Operator
30
2.12: Divergence and Curl
30
2.13: Second Derivatives and Laplace Operator
30
2.14: Line, Surface, and Volume Integrals
30
2.15: Divergence and Stokes' Theorems

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: Apparent Weight
30
5.10: Newton's Third Law: Introduction
30
5.11: Newton's Third Law: Examples
30
5.12: Drawing Free-body Diagrams: Rules
30
5.13: Free Body Diagrams: Examples
30
5.14: Inertial Frames of Reference
30
5.15: Non-inertial Frames of Reference
30
5.16: Centrifugal Force

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
30
6.9: Drag Force and Terminal Speed
30
6.10: Tension

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: Work and Energy for Variable Forces
30
7.10: Work-Energy Theorem for Motion Along a Curve
30
7.11: Work Done Over an Inclined Plane
30
7.12: Work Done by Many Forces
30
7.13: Work Done by Gravity
30
7.14: Power
30
7.15: 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: Conservation of Mechanical Energy
30
8.4: Work Done on a System by External Force
30
8.5: Conservative Forces
30
8.6: Non-conservative Forces
30
8.7: Conservation of Energy
30
8.8: Conservation of Energy: Application
30
8.9: Force and Potential Energy in One Dimension
30
8.10: Force and Potential Energy in Three Dimensions
30
8.11: Energy Diagrams - I
30
8.12: 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: Gravitational Potential Energy for Extended Objects
30
9.16: Rocket Propulsion in Empty Space - I
30
9.17: Rocket Propulsion In Empty Space - II
30
9.18: Rocket Propulsion in Gravitational Field - I
30
9.19: 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: Moment of Inertia of Compound Objects
30
10.11: Parallel-axis Theorem
30
10.12: Perpendicular-Axis Theorem
30
10.13: Vector Transformation in Rotating Coordinate Systems
30
10.14: Coriolis Force

Chapter 11: Dynamics of Rotational Motions

30
11.1: Torque
30
11.2: Net Torque Calculations
30
11.3: Equation of Rotational Dynamics
30
11.4: Rolling Without Slipping
30
11.5: Rolling With Slipping
30
11.6: Work and Power for Rotational Motion
30
11.7: Work-Energy Theorem for Rotational Motion
30
11.8: Angular Momentum: Single Particle
30
11.9: Angular Momentum: Rigid Body
30
11.10: Conservation of Angular Momentum
30
11.11: Conservation of Angular Momentum: Application
30
11.12: Rotation of Asymmetric Top
30
11.13: Gyroscope
30
11.14: 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: Problem Solving on Stress and Strain
30
12.10: Indeterminate Structure
30
12.11: Elasticity
30
12.12: Plasticity

Chapter 13: Fluid Mechanics

30
13.1: Characteristics of Fluids
30
13.2: Density
30
13.3: Pressure of Fluids
30
13.4: Variation of Atmospheric Pressure
30
13.5: Pascal's Law
30
13.6: Application of Pascal's Law
30
13.7: Pressure Gauges
30
13.8: Buoyancy
30
13.9: Archimedes' Principle
30
13.10: Density and Archimedes' Principle
30
13.11: Accelerating Fluids
30
13.12: Surface Tension and Surface Energy
30
13.13: Excess Pressure Inside a Drop and a Bubble
30
13.14: Contact Angle
30
13.15: Rise of Liquid in a Capillary Tube
30
13.16: Laminar and Turbulent Flow
30
13.17: Equation of Continuity
30
13.18: Bernoulli's Equation
30
13.19: Bernoulli's Principle
30
13.20: Bernoulli's Principle: Applications
30
13.21: Energy Conservation and Bernoulli's Equation
30
13.22: Viscosity
30
13.23: Poiseuille's Law and Reynolds Number
30
13.24: Stokes' Law

Chapter 14: Gravitation

30
14.1: Gravitation
30
14.2: Newton's Law of Gravitation
30
14.3: Gravitation Between Spherically Symmetric Masses
30
14.4: Gravity between Spherical Bodies
30
14.5: Reduced Mass Coordinates: Isolated Two-body Problem
30
14.6: Acceleration due to Gravity on Earth
30
14.7: Acceleration due to Gravity on Other Planets
30
14.8: Apparent Weight and the Earth's Rotation
30
14.9: Variation in Acceleration due to Gravity near the Earth's Surface
30
14.10: Potential Energy due to Gravitation
30
14.11: The Principle of Superposition and the Gravitational Field
30
14.12: Escape Velocity
30
14.13: Circular Orbits and Critical Velocity for Satellites
30
14.14: Energy of a Satellite in a Circular Orbit
30
14.15: Kepler's First Law of Planetary Motion
30
14.16: Kepler's Second Law of Planetary Motion
30
14.17: Kepler's Third Law of Planetary Motion
30
14.18: Tidal Forces
30
14.19: Schwarzschild Radius and Event Horizon
30
14.20: Detection of Black Holes
30
14.21: Principle of Equivalence
30
14.22: 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: Frequency of Spring-Mass System
30
15.6: Simple Harmonic Motion and Uniform Circular Motion
30
15.7: Problem Solving: Energy in Simple Harmonic Motion
30
15.8: Simple Pendulum
30
15.9: Torsional Pendulum
30
15.10: Physical Pendulum
30
15.11: Measuring Acceleration Due to Gravity
30
15.12: Damped Oscillations
30
15.13: Types of Damping
30
15.14: Forced Oscillations
30
15.15: 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: Graphing the Wave Function
30
16.5: Velocity and Acceleration of a Wave
30
16.6: Speed of a Transverse Wave
30
16.7: Problem-Solving: Tuning of a Guitar String
30
16.8: Kinetic and Potential Energy of a Wave
30
16.9: Energy and Power of a Wave
30
16.10: Interference and Superposition of Waves
30
16.11: Reflection of Waves
30
16.12: Propagation of Waves
30
16.13: Standing Waves
30
16.14: Modes of Standing Waves - I
30
16.15: Modes of Standing Waves: II

Chapter 17: Sound

30
17.1: Sound Waves
30
17.2: Sound as Pressure Waves
30
17.3: Perception of Sound Waves
30
17.4: Speed of Sound in Solids and Liquids
30
17.5: Speed of Sound in Gases
30
17.6: Deriving the Speed of Sound in a Liquid
30
17.7: Sound Intensity
30
17.8: Sound Intensity Level
30
17.9: Intensity and Pressure of Sound Waves
30
17.10: Sound Waves: Interference
30
17.11: Interference: Path Lengths
30
17.12: Sound Waves: Resonance
30
17.13: Beats
30
17.14: Doppler Effect - I
30
17.15: Doppler Effect - II
30
17.16: Shock Waves
30
17.17: Echo

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: Thermal expansion and Thermal stress: Problem Solving
30
18.8: Heat Flow and Specific Heat
30
18.9: Phase Changes
30
18.10: Calorimetry
30
18.11: Mechanisms of Heat Transfer I
30
18.12: Mechanisms of Heat Transfer II
30
18.13: Conduction, Convection and Radiation: Problem Solving
30
18.14: Radiation: Applications
30
18.15: Absorption of Radiation

Chapter 19: The Kinetic Theory of Gases

30
19.1: Equation of State
30
19.2: Ideal Gas Equation
30
19.3: Van der Waals Equation
30
19.4: pV-Diagrams
30
19.5: Kinetic Theory of an Ideal Gas
30
19.6: Molecular Kinetic Energy
30
19.7: Distribution of Molecular Speeds
30
19.8: Maxwell-Boltzmann Distribution: Problem Solving
30
19.9: Phase Diagram
30
19.10: Mean free path and Mean free time
30
19.11: Heat Capacity: Problem-Solving
30
19.12: Dalton's Law of Partial Pressure
30
19.13: Escape Velocities of Gases

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: Heat and Free Expansion
30
20.5: Internal Energy
30
20.6: First Law of Thermodynamics
30
20.7: First Law Of Thermodynamics: Problem-Solving
30
20.8: Cyclic Processes And Isolated Systems
30
20.9: Isothermal Processes
30
20.10: Isochoric and Isobaric Processes
30
20.11: Heat Capacities of an Ideal Gas I
30
20.12: Heat Capacities of an Ideal Gas II
30
20.13: Heat Capacities of an Ideal Gas III
30
20.14: Adiabatic Processes for an Ideal Gas
30
20.15: Pressure and Volume in an Adiabatic Process
30
20.16: Work Done in an Adiabatic Process
30
20.17: Thermodynamic Potentials
30
20.18: Maxwell's Thermodynamic Relations
30
20.19: Joule-Thomson Effect

Chapter 21: The Second Law of Thermodynamics

30
21.1: Reversible and Irreversible Processes
30
21.2: Heat Engines
30
21.3: Internal Combustion Engine
30
21.4: Otto and Diesel Cycle
30
21.5: Refrigerators and Heat Pumps
30
21.6: Statements of the Second Law of Thermodynamics
30
21.7: The Carnot Cycle
30
21.8: Efficiency of The Carnot Cycle
30
21.9: The Carnot Cycle and the Second Law of Thermodynamics
30
21.10: Entropy
30
21.11: Entropy Change in Reversible Processes
30
21.12: 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: Comparison Between Electrical And Gravitational Forces
30
22.8: Electric Field
30
22.9: Electric Field of Two Equal and Opposite Charges
30
22.10: Continuous Charge Distributions
30
22.11: Electric Field of a Continuous Line Charge
30
22.12: Electric Field of a Charged Disk
30
22.13: Electric Field Lines
30
22.14: Properties of Electric Field Lines
30
22.15: Electric Dipoles and Dipole Moment
30
22.16: Induced Electric Dipoles

Chapter 23: Gauss's Law

30
23.1: Electric Flux
30
23.2: Calculation of Electric Flux
30
23.3: Gauss's Law
30
23.4: Gauss's Law: Problem-Solving
30
23.5: Gauss's Law: Spherical Symmetry
30
23.6: Gauss's Law: Cylindrical Symmetry
30
23.7: Gauss's Law: Planar Symmetry
30
23.8: Electric Field Inside a Conductor
30
23.9: Charge on a Conductor
30
23.10: Electric Field at the Surface of a Conductor
30
23.11: Electric Field of a Non Uniformly Charged Sphere
30
23.12: Electric Field of Parallel Conducting Plates
30
23.13: Divergence and Curl of Electric Field

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: Calculations of Electric Potential I
30
24.7: Calculations of Electric Potential II
30
24.8: Equipotential Surfaces and Field Lines
30
24.9: Equipotential Surfaces and Conductors
30
24.10: Determining Electric Field From Electric Potential
30
24.11: Poisson's And Laplace's Equation
30
24.12: Van de Graaff Generator
30
24.13: Energy Associated With a Charge Distribution
30
24.14: Electrostatic Boundary Conditions
30
24.15: Second Uniqueness Theorem

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: Equivalent Capacitance
30
25.5: Energy Stored in a Capacitor
30
25.6: Energy Stored in a Capacitor: Problem Solving
30
25.7: Capacitor With A Dielectric
30
25.8: Dielectric Polarization in a Capacitor
30
25.9: Gauss's Law in Dielectrics
30
25.10: Potential Due to a Polarized Object
30
25.11: Susceptibility, Permittivity and Dielectric Constant
30
25.12: Electrostatic Boundary Conditions in Dielectrics

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: Non-ohmic Devices
30
26.8: Electrical Power
30
26.9: Electrical Energy
30
26.10: Continuity Equation
30
26.11: Boundary Conditions for Current Density
30
26.12: Electrical Conductivity
30
26.13: Theory of Metallic Conduction

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: Combination Of Resistors
30
27.5: Kirchhoff's Rules
30
27.6: Kirchoff's Rules: Application
30
27.7: DC Battery
30
27.8: Multiple Voltage Sources
30
27.9: Galvanometer
30
27.10: Ammeter
30
27.11: Voltmeter
30
27.12: Potentiometer
30
27.13: Wheatstone Bridge
30
27.14: Power Dissipated in a Circuit: Problem Solving
30
27.15: RC Circuits: Charging A Capacitor
30
27.16: RC Circuits: Discharging A Capacitor
30
27.17: Applications of RC Circuits
30
27.18: Household Wiring And Electrical Safety

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
30
28.7: Magnetic Force On A Current-Carrying Conductor
30
28.8: Magnetic Force On Current-Carrying Wires: Example
30
28.9: Force On A Current Loop In A Magnetic Field
30
28.10: Torque On A Current Loop In A Magnetic Field
30
28.11: The Hall Effect
30
28.12: Thomson's e/m Experiment

Chapter 29: Sources of Magnetic Fields

30
29.1: Magnetic Field due to Moving Charges
30
29.2: Biot-Savart Law
30
29.3: Biot-Savart Law: Problem-Solving
30
29.4: Magnetic Field Due To A Thin Straight Wire
30
29.5: Magnetic Field Due to Two Straight Wires
30
29.6: Magnetic Force Between Two Parallel Currents
30
29.7: Magnetic Field Of A Current Loop
30
29.8: Divergence and Curl of Magnetic Field
30
29.9: Ampere's Law
30
29.10: Ampere's Law: Problem-Solving
30
29.11: Solenoids
30
29.12: Magnetic Field of a Solenoid
30
29.13: Toroids
30
29.14: Magnetic Vector Potential
30
29.15: Potential Due to a Magnetized Object
30
29.16: Magnetostatic Boundary Conditions
30
29.17: Ampere's Law in Matter
30
29.18: Magnetic Susceptibility and Permeability
30
29.19: Magnetic Moment of an Electron
30
29.20: Diamagnetism
30
29.21: Paramagnetism
30
29.22: 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: Faraday Disk Dynamo
30
30.6: Induced Electric Fields
30
30.7: Induced Electric Fields: Applications
30
30.8: Eddy Currents
30
30.9: Displacement Current
30
30.10: Significance of Displacement Current
30
30.11: Electromagnetic Fields
30
30.12: Maxwell's Equation Of Electromagnetism
30
30.13: Symmetry in Maxwell's Equations
30
30.14: Ampere-Maxwell's Law: Problem-Solving
30
30.15: Differential Form of Maxwell's Equations
30
30.16: Electric Generator: Alternator
30
30.17: DC Generator
30
30.18: Back EMF
30
30.19: Magnetic Damping
30
30.20: Superconductor
30
30.21: Types Of Superconductors

Chapter 31: Inductance

30
31.1: Mutual Inductance
30
31.2: Self-Inductance
30
31.3: Calculation of Self-inductance
30
31.4: Inductors
30
31.5: Energy In A Magnetic Field
30
31.6: Energy Stored In A Coaxial Cable
30
31.7: RL Circuits
30
31.8: Current Growth And Decay In RL Circuits
30
31.9: Comparison between RL and RC circuits
30
31.10: LC Circuits
30
31.11: Oscillations In An LC Circuit
30
31.12: RLC Series Circuits
30
31.13: RLC Circuit as a Damped Oscillator

Chapter 32: Alternating-Current Circuits

30
32.1: AC Sources
30
32.2: RMS Value in AC Circuit
30
32.3: Resistor in an AC Circuit
30
32.4: Capacitor in an AC Circuit
30
32.5: Inductor in an AC Circuit
30
32.6: RLC Series Circuits: Introduction
30
32.7: RLC Series Circuits: Impedance
30
32.8: RLC Series Circuit: Problem-Solving
30
32.9: Power in an AC Circuit
30
32.10: Resonance in an AC Circuit
30
32.11: Transformers
30
32.12: Types Of Transformers
30
32.13: Energy Losses in Transformers

Chapter 33: Electromagnetic Waves

30
33.1: Electromagnetic Waves
30
33.2: Generating Electromagnetic Radiations
30
33.3: The Electromagnetic Spectrum
30
33.4: Electromagnetic Wave Equation
30
33.5: Plane Electromagnetic Waves I
30
33.6: Plane Electromagnetic Waves II
30
33.7: Propagation Speed of Electromagnetic Waves
30
33.8: Electromagnetic Waves in Matter
30
33.9: Energy Carried By Electromagnetic Waves
30
33.10: Intensity Of Electromagnetic Waves
30
33.11: Momentum And Radiation Pressure
30
33.12: Radiation Pressure: Problem Solving
30
33.13: Standing Electromagnetic Waves
30
33.14: Standing Waves in a Cavity

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TRANSCRIPT

A parallel plate capacitor in vacuum is in the charging condition. The radius of each plate is 5 cm. At an instant, the conduction current flowing through the wires is 0.4 A. What will be the displacement current between the plates at this time?

Also, calculate the displacement current density and evaluate the rate of change in the electric field between the plates. Calculate the induced magnetic field between the plates and plot it for a distance of 1--5 cm from the axis.

Displacement current is expressed in terms of electric flux, which in turn, can be obtained in terms of charge.

The displacement current equals the conduction current flowing through the wires, which is 0.4 Amperes.

Furthermore, the displacement current density can be calculated.

As the capacitor charges, the electric field between the plates increases, which is evaluated using displacement current density and the permittivity of free space.

As per the Ampere-Maxwell law, the magnetic field is induced between the capacitor plates, which is calculated for different distances from the axis and plotted as shown.

30.14: Ampere-Maxwell's Law: Problem-Solving

A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?

To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.

For the first part of the problem, the voltage between the plates at time t is given by

Equation1

Suppose that the z-axis points from the positive plate to the negative plate. In this case, the z-component of the electric field between the plates as a function of time t is given by

Equation2

Therefore, the z-component of the displacement current I_d between the plates of the capacitor can be evaluated as

Equation3

where the capacitance Equation 4 is being used.

Further, to solve the second part of the above-stated problem, the current into the capacitor after the circuit is closed can be obtained by the charge on the capacitor, which in turn can be evaluated by using the expression for V_C. As a result, the real current into the capacitor is found to be the same as the displacement current.

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

30.14: Ampere-Maxwell's Law: Problem-Solving

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