Electromagnetic Fields
What is Electromagnetic Fields?
Force per unit positive test charge (N/C or V/m).
Key formula / rule: Gauss's Law for Electric Fields (Differential Form)
Key points
- Understand and apply Maxwell's equations in differential and integral forms.
- Calculate electric and magnetic fields for various charge and current distributions.
- Apply boundary conditions to solve problems involving different media.
- Analyze the propagation of electromagnetic waves in lossless and lossy media.
Common exam trap
Incorrect application of vector calculus operators (gradient, divergence, curl).
Definitions
- Term
Electric Field Intensity (E)
- Meaning
Force per unit positive test charge (N/C or V/m).
- Term
Electric Flux Density (D)
- Meaning
Electric field intensity multiplied by permittivity (C/m²).
- Term
Magnetic Field Intensity (H)
- Meaning
Magnetic field strength produced by currents (A/m).
- Term
Magnetic Flux Density (B)
- Meaning
Magnetic field intensity multiplied by permeability (Tesla or Wb/m²).
- Term
Permittivity (ε)
- Meaning
A measure of how an electric field affects, and is affected by, a dielectric medium (F/m).
- Term
Permeability (μ)
- Meaning
A measure of the ability of a material to support the formation of a magnetic field within itself (H/m).
- Term
Displacement Current
- Meaning
The term ∂D/∂t in Ampere-Maxwell's law, representing a time-varying electric flux that produces a magnetic field, analogous to a conduction current.
- Term
Poynting Vector
- Meaning
A vector whose direction is the direction of energy flow of an electromagnetic field and whose magnitude is the power per unit area (W/m²).
- Term
Lorentz Force
- Meaning
The total force exerted by an electromagnetic field on a charged particle.
Learning objectives
Understand and apply Maxwell's equations in differential and integral forms.
Calculate electric and magnetic fields for various charge and current distributions.
Apply boundary conditions to solve problems involving different media.
Analyze the propagation of electromagnetic waves in lossless and lossy media.
Calculate electromagnetic power flow using the Poynting vector.
Understand the Lorentz force and its applications.
Formulae
- Name
Gauss's Law for Electric Fields (Differential Form)
- Note
Relates electric flux density to volume charge density.
- Expression
∇ ⋅ D = ρv
- Name
Gauss's Law for Electric Fields (Integral Form)
- Note
Total electric flux through a closed surface equals the enclosed charge.
- Expression
∮S D ⋅ dS = Qenc
- Name
Gauss's Law for Magnetic Fields (Differential Form)
- Note
Magnetic monopoles do not exist; magnetic field lines are always closed loops.
- Expression
∇ ⋅ B = 0
- Name
Gauss's Law for Magnetic Fields (Integral Form)
- Note
Total magnetic flux through any closed surface is zero.
- Expression
∮S B ⋅ dS = 0
- Name
Faraday's Law of Induction (Differential Form)
- Note
A time-varying magnetic field induces an electric field (electromotive force).
- Expression
∇ × E = -∂B/∂t
- Name
Faraday's Law of Induction (Integral Form)
- Note
The electromotive force around a closed loop equals the negative time rate of change of magnetic flux through the surface bounded by the loop.
- Expression
∮C E ⋅ dL = -∫S (∂B/∂t) ⋅ dS
- Name
Ampere-Maxwell Law (Differential Form)
- Note
Relates magnetic field intensity to conduction current density and displacement current density.
- Expression
∇ × H = J + ∂D/∂t
- Name
Ampere-Maxwell Law (Integral Form)
- Note
The magnetomotive force around a closed loop equals the sum of enclosed conduction current and displacement current.
- Expression
∮C H ⋅ dL = Ienc + ∫S (∂D/∂t) ⋅ dS
- Name
Constitutive Relation for Electric Fields
- Note
Relates electric flux density (D) to electric field intensity (E) via permittivity (ε = ε₀εᵣ).
- Expression
D = εE
- Name
Constitutive Relation for Magnetic Fields
- Note
Relates magnetic flux density (B) to magnetic field intensity (H) via permeability (μ = μ₀μᵣ).
- Expression
B = μH
- Name
Ohm's Law (Point Form)
- Note
Relates conduction current density (J) to electric field intensity (E) via conductivity (σ).
- Expression
J = σE
- Name
Lorentz Force Law
- Note
Force experienced by a charge 'q' moving with velocity 'v' in electric field 'E' and magnetic field 'B'.
- Expression
F = q(E + v × B)
- Name
Poynting Vector
- Note
Represents the instantaneous power flow density (W/m²) and direction of electromagnetic energy.
- Expression
P = E × H
- Name
Continuity Equation
- Note
Statement of charge conservation, relating current density to the time rate of change of charge density.
- Expression
∇ ⋅ J = -∂ρv/∂t
- Name
Wave Equation for Electric Field (source-free, lossless)
- Note
Describes the propagation of electric field in a medium.
- Expression
∇²E - με (∂²E/∂t²) = 0
- Name
Wave Equation for Magnetic Field (source-free, lossless)
- Note
Describes the propagation of magnetic field in a medium.
- Expression
∇²H - με (∂²H/∂t²) = 0
- Name
Speed of Light in a Medium
- Note
Velocity of electromagnetic wave propagation in a medium with permittivity ε and permeability μ.
- Expression
v = 1/√(με)
- Name
Intrinsic Impedance of a Medium
- Note
Ratio of electric field to magnetic field magnitudes for a plane wave in a medium.
- Expression
η = √(μ/ε)
- Name
Coulomb's Law
- Note
Force between two point charges q₁ and q₂ separated by distance R.
- Expression
F = (1/(4πε₀)) * (q₁q₂/R²) * aR
- Name
Electric Field Intensity (point charge)
- Note
Electric field produced by a point charge q at distance R.
- Expression
E = (1/(4πε₀)) * (q/R²) * aR
- Name
Electric Potential
- Note
Potential difference is the negative line integral of electric field.
- Expression
V = -∫ E ⋅ dL
- Name
Poisson's Equation
- Note
Relates electric potential to volume charge density.
- Expression
∇²V = -ρv/ε
- Name
Laplace's Equation
- Note
Poisson's equation in a charge-free region.
- Expression
∇²V = 0
- Name
Biot-Savart Law
- Note
Differential magnetic field intensity produced by a differential current element.
- Expression
dH = (I dL × aR) / (4πR²)
- Name
Magnetic Flux
- Note
Total magnetic flux passing through a surface S.
- Expression
Φ = ∫S B ⋅ dS
- Name
Magnetic Vector Potential
- Note
Magnetic flux density can be expressed as the curl of the magnetic vector potential A.
- Expression
B = ∇ × A
Prerequisites
Vector Calculus (gradient, divergence, curl, line, surface, and volume integrals, divergence theorem, Stokes' theorem)
Basic Physics (Coulomb's Law, Biot-Savart Law, Ohm's Law)
Differential Equations
Basic knowledge of electric circuits
Common mistakes
Incorrect application of vector calculus operators (gradient, divergence, curl).
Confusing electric field intensity (E) with electric flux density (D), and magnetic field intensity (H) with magnetic flux density (B).
Sign errors in Faraday's Law or Lorentz Force.
Incorrectly applying boundary conditions at material interfaces.
Not understanding the physical significance of each term in Maxwell's equations.
Errors in unit conversions or using incorrect fundamental constants.
Keywords
Maxwell's Equations
Electromagnetism
Electric Field
Magnetic Field
Poynting Vector
Lorentz Force
Wave Propagation
Electrostatics
Magnetostatics
Permittivity
Permeability
Displacement Current
Boundary Conditions
Vector Calculus
Practice preview
Which of the following is the unit of magnetic flux density?…
easy
A circular loop of radius R carries a current I. What is the magnetic field at the center of the loop?…
hard
The electric field intensity at a point due to a point charge is given by E = kq/r². What is the unit of k?…
easy