Maxwell's Equations
What is Maxwell's Equations?
A vector field that represents the effect of electric fields in materials, accounting for free charges and polarization.
Key formula / rule: Gauss's Law for Electricity (Differential Form)
Key points
- Understand the physical meaning of each of Maxwell's equations.
- Be able to write Maxwell's equations in both differential and integral forms.
- Apply Maxwell's equations to solve problems in electrostatics, magnetostatics, and electrodynamics.
- Understand the concept of displacement current and its role.
Common exam trap
Confusing differential and integral forms.
Definitions
- Term
Electric Displacement Field (D)
- Meaning
A vector field that represents the effect of electric fields in materials, accounting for free charges and polarization.
- Term
Magnetic Flux Density (B)
- Meaning
A vector field that describes the strength and direction of a magnetic field, representing the force on a moving charge.
- Term
Electric Field Intensity (E)
- Meaning
A vector field representing the force per unit charge experienced by a test charge.
- Term
Magnetic Field Intensity (H)
- Meaning
A vector field that represents the magnetizing effect of electric currents and magnetic materials.
- Term
Charge Density (ρv)
- Meaning
The amount of electric charge per unit volume.
- Term
Current Density (J)
- Meaning
The amount of electric current per unit cross-sectional area.
- Term
Displacement Current
- Meaning
A term added by Maxwell to Ampère's law, representing the effect of a time-varying electric field as a source of magnetic field, analogous to conduction current.
Learning objectives
Understand the physical meaning of each of Maxwell's equations.
Be able to write Maxwell's equations in both differential and integral forms.
Apply Maxwell's equations to solve problems in electrostatics, magnetostatics, and electrodynamics.
Understand the concept of displacement current and its role.
Recognize the derivation of electromagnetic wave equations from Maxwell's equations.
Formulae
- Name
Gauss's Law for Electricity (Differential Form)
- Note
Relates the divergence of the electric displacement field (D) to the volume charge density (ρv).
- Expression
\nabla \· \mathbf{D} = \ρv
- Name
Gauss's Law for Magnetism (Differential Form)
- Note
States that the divergence of the magnetic flux density (B) is always zero, implying no magnetic monopoles.
- Expression
\nabla \· \mathbf{B} = 0
- Name
Faraday's Law of Induction (Differential Form)
- Note
Relates the curl of the electric field (E) to the time rate of change of the magnetic flux density (B).
- Expression
\nabla \× \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
- Name
Ampère-Maxwell Law (Differential Form)
- Note
Relates the curl of the magnetic field intensity (H) to the current density (J) and the time rate of change of the electric displacement field (D), known as displacement current.
- Expression
\nabla \× \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}
- Name
Gauss's Law for Electricity (Integral Form)
- Note
The total electric flux through a closed surface is equal to the total charge enclosed by the surface.
- Expression
\ointS \mathbf{D} \· d\mathbf{S} = Qenc
- Name
Gauss's Law for Magnetism (Integral Form)
- Note
The total magnetic flux through any closed surface is zero.
- Expression
\ointS \mathbf{B} \· d\mathbf{S} = 0
- Name
Faraday's Law of Induction (Integral Form)
- Note
The electromotive force (EMF) around a closed circuit is equal to the negative of the time rate of change of magnetic flux through the surface bounded by the circuit.
- Expression
\ointC \mathbf{E} \· d\mathbf{l} = -\intS \frac{\partial \mathbf{B}}{\partial t} \· d\mathbf{S}
- Name
Ampère-Maxwell Law (Integral Form)
- Note
The line integral of the magnetic field intensity around a closed loop is equal to the total current (conduction + displacement) passing through the surface bounded by the loop.
- Expression
\ointC \mathbf{H} \· d\mathbf{l} = \intS (\mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}) \· d\mathbf{S}
- Name
Constitutive Relation (Linear, Isotropic)
- Note
Relates electric displacement field (D) to electric field (E) via permittivity (ε).
- Expression
\mathbf{D} = \ε \mathbf{E}
- Name
Constitutive Relation (Linear, Isotropic)
- Note
Relates magnetic flux density (B) to magnetic field intensity (H) via permeability (μ).
- Expression
\mathbf{B} = \μ \mathbf{H}
- Name
Constitutive Relation (Linear, Isotropic)
- Note
Relates conduction current density (J) to electric field (E) via conductivity (σ) (Ohm's Law in point form).
- Expression
\mathbf{J} = \σ \mathbf{E}
Prerequisites
Vector Calculus (gradient, divergence, curl, line integrals, surface integrals, volume integrals).
Basic concepts of Electric and Magnetic Fields (charge, current, flux, potential).
Understanding of differential equations.
Common mistakes
Confusing differential and integral forms.
Incorrectly applying boundary conditions.
Forgetting the displacement current term in Ampère's law.
Misinterpreting the physical significance of each equation.
Keywords
Maxwell's Equations
Gauss's Law
Faraday's Law
Ampère's Law
Displacement Current
Electromagnetism
Electromagnetic Waves
Vector Calculus
Electric Field
Magnetic Field
Practice preview
What does Gauss's Law for electric fields state in its integral form?…
easy
Which of Maxwell's equations implies the non-existence of magnetic monopoles?…
easy
Which term in the Ampere-Maxwell law accounts for the generation of a magnetic field by a time-varying electric field?…
medium