Electromagnetic Fields and Maxwell’s Equations

Introduction and context

Electromagnetic theory explains how electric charges, currents, materials, and time-varying fields interact. It provides the foundation for electrical power systems, motors and generators, antennas, optical communication, radar, wireless networks, electromagnetic compatibility, sensors, and semiconductor devices.

The theory developed from several experimentally established laws. Coulomb’s law describes electrostatic force, Gauss’s law relates electric flux to enclosed charge, Ampère’s law relates magnetic fields to current, Faraday’s law describes electromagnetic induction, and the Lorentz force gives the force on a charged particle. Maxwell showed that these laws form a unified system and introduced the displacement-current term necessary for consistency with charge conservation.

At the bachelor level, the central task is to understand both the physical meaning and the mathematical structure of fields. A field is a quantity defined at every point in space and time. The electric field E describes electric force per unit charge, while the magnetic flux density B describes magnetic effects associated with moving charges and currents.

Learning objectives

After studying this chapter, a student should be able to:

• distinguish electric field, electric flux density, magnetic field, and magnetic flux density;
• apply Gauss’s law to electrostatic systems with suitable symmetry;
• explain polarization and electric displacement in dielectric materials;
• apply Ampère’s law and the Biot–Savart law to magnetostatic systems;
• explain Faraday’s law, Lenz’s law, and electromagnetic induction;
• justify Maxwell’s displacement-current correction;
• state and interpret Maxwell’s equations in integral and differential forms;
• apply electromagnetic boundary conditions at material interfaces;
• calculate electromagnetic energy density and power flow using the Poynting vector;
• evaluate a practical electromagnetic system and identify modelling assumptions.

1. Fundamental field quantities

1.1 Electric field and electric flux density

The electric field at a point is defined as the force per unit positive test charge:

Here, E is electric field intensity in newtons per coulomb (N/C), equivalently volts per metre (V/m), F is force in newtons, and q is charge in coulombs. The test charge must be sufficiently small that it does not disturb the source distribution.

For a point charge Q in free space, Coulomb’s law gives

where r is distance from the charge, r̂ is the outward radial unit vector, and ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space. This expression applies to a stationary point charge in an electrostatic, homogeneous, isotropic free-space environment.

Electric flux density is defined by

where D is measured in C/m² and P is polarization, measured in C/m². In a linear, homogeneous, isotropic dielectric,

where ε is absolute permittivity in F/m and εᵣ is relative permittivity, a dimensionless material property.

1.2 Magnetic quantities

The magnetic flux density B is measured in tesla (T), equivalent to Wb/m². The magnetic field intensity H is measured in amperes per metre (A/m). In a linear, isotropic material,

where μ is permeability in H/m, μ₀ = 4π × 10⁻⁷ H/m is free-space permeability, and μᵣ is relative permeability.

Unlike electric field lines, magnetic field lines form closed loops because isolated magnetic monopoles have not been observed. Consequently, the net magnetic flux through any closed surface is zero.

1.3 Force on charges and currents

The Lorentz force on a particle of charge q moving with velocity v is

The electric contribution may act on a stationary charge, whereas the magnetic contribution is perpendicular to both velocity and magnetic flux density. Thus, magnetic force changes the direction of motion but does no instantaneous work on an isolated point charge because v · (v × B) = 0.

For a current-carrying differential conductor,

where I is current in amperes and dl is a directed differential length vector.

2. Electrostatics and Gauss’s law

Electrostatics concerns charges at rest and fields that do not vary with time. In electrostatics,

This means that the electric field is conservative and can be expressed as the negative gradient of electric potential V:

The potential difference between points a and b is

The line integral is path-independent in an electrostatic field.

Gauss’s law in integral form is

where the closed surface S encloses free charge Q_f,enc and da is an outward-directed area vector. Its differential form is

where ρ_f is free-charge volume density in C/m³. Gauss’s law is always valid, but it is especially useful when symmetry permits D to be taken as constant over a Gaussian surface.

Example 1: Electric field of a uniformly charged sphere

Consider a solid sphere of radius a carrying total uniformly distributed free charge Q. For r ≥ a, spherical symmetry gives constant radial flux density over a spherical Gaussian surface:

Therefore,

and in free space,

For r < a, the enclosed charge is Q(r³/a³), so

The interior field increases linearly with r, while the exterior field decreases as 1/r². The result depends critically on uniform spherical symmetry; Gauss’s law alone does not make arbitrary charge distributions easy to solve.

3. Polarization and dielectric materials

In an insulating material, an applied electric field slightly separates positive and negative charges within atoms or molecules. These microscopic electric dipoles produce polarization P, the electric dipole moment per unit volume.

For a linear isotropic dielectric,

where χₑ is electric susceptibility. Since

we obtain

Thus, εᵣ = 1 + χₑ for this ideal linear material model.

Polarization produces bound charge. The bound volume-charge density is

and the bound surface-charge density is

where n̂ is the outward unit normal from the dielectric. Free charge is supplied by conductors or external sources; bound charge results from material polarization.

At a conductor in electrostatic equilibrium, the electric field inside the conductor is zero. Excess free charge resides on its surface, and the conductor surface is an equipotential. Just outside the conductor,

and

where E_t is tangential electric field, D_n is normal electric flux density, and ρ_s is free surface-charge density.

4. Magnetostatics

Magnetostatics concerns steady currents and time-independent magnetic fields. The magnetic field generated by a current element is described by the Biot–Savart law:

This free-space expression applies to steady current distributions. For a long straight conductor carrying current I, integration gives

where ρ is perpendicular distance from the conductor and φ̂ is the azimuthal direction determined by the right-hand rule.

Ampère’s law for magnetostatics is

or, in differential form,

where I_f,enc is enclosed free current and J_f is free-current density in A/m². For a long straight wire in free space, H = I/(2πρ), and B = μ₀H.

The divergence law for magnetism is

It states that magnetic flux has no beginning or end. In integral form,

5. Electromagnetic induction

Faraday’s law states that a time-varying magnetic flux produces an electromotive force (emf):

where Φ_B = ∫_S B · da is magnetic flux in webers and ℰ is induced emf in volts. The negative sign expresses Lenz’s law: the induced effect opposes the change in magnetic flux that produces it.

For a stationary loop with a time-varying field, the differential form is

The more general flux rule for a moving conducting loop includes motional emf:

where v is the local velocity of the conductor. Transformer emf is associated primarily with time-varying B, whereas motional emf arises when a conductor moves through a magnetic field.

6. Maxwell’s correction and displacement current

The magnetostatic form of Ampère’s law, ∇ × H = J, creates a difficulty for time-varying systems. Taking divergence of both sides gives

The left side is identically zero, but charge conservation requires

These are inconsistent when charge density changes with time. From Gauss’s law, ρ = ∇ · D, so

Therefore, the corrected Ampère–Maxwell law is

The term ∂D/∂t is called displacement current density. It is not necessarily a motion of charged particles through a material; it represents the electromagnetic effect of a changing electric flux. In integral form,

This correction allows current continuity through a capacitor gap. Conduction current flows in the wires, while changing electric flux between the plates provides the corresponding displacement-current term.

7. Maxwell’s equations

In macroscopic form, Maxwell’s equations are:

The integral forms are useful for finite surfaces, circuits, and symmetry arguments. The differential forms describe local field behaviour and are useful in field computation and wave analysis. The divergence theorem connects flux integrals with divergence; Stokes’ theorem connects circulation integrals with curl.

For a linear, isotropic, homogeneous medium, the constitutive relations are

where σ is conductivity in S/m. These relations are additional material models, not replacements for Maxwell’s equations. In anisotropic media, ε, μ, or σ may be tensors rather than scalars.

8. Boundary conditions

Boundary conditions determine how fields change at an interface between two media. Let n̂ point from medium 1 to medium 2, and let ρ_s be free surface-charge density.

The normal component of D satisfies

This follows from a thin Gaussian pillbox. The normal component of B is continuous:

This follows from the absence of magnetic monopoles.

For a stationary interface, the tangential component of E satisfies

unless singular time-varying conditions are present. The tangential component of H satisfies

where K_f is free surface-current density in A/m.

At a dielectric interface with no free surface charge, D₁n = D₂n. If ε₁ and ε₂ differ, then E₁n and E₂n generally differ. With no surface current, H₁t = H₂t; since B = μH, tangential B may differ when permeabilities differ.

9. Electromagnetic energy and the Poynting vector

Electric and magnetic fields store energy. In a linear, nondispersive medium, the electromagnetic energy density is

For a linear isotropic medium this becomes

where E and H denote field magnitudes. The units are joules per cubic metre.

The Poynting vector is

and has units W/m². Its direction represents instantaneous electromagnetic power flow. The integral of S over a surface gives power crossing that surface:

Poynting’s theorem expresses conservation of electromagnetic energy:

The three terms represent outward power flow, rate of increase of stored field energy, and power delivered to matter. If J · E is positive, electromagnetic energy is converted into material energy, commonly heat.

In a lossless uniform plane wave, electric and magnetic fields are perpendicular, and the time-average power flow is often written

for phasor fields, where Re denotes the real part and * denotes complex conjugate. This phasor expression assumes sinusoidal steady state.

Worked example 2: Power flow in a plane wave

A uniform plane wave in free space has electric-field peak magnitude E₀ = 120 V/m. The intrinsic impedance of free space is η₀ ≈ 377 Ω. For a sinusoidal plane wave, H₀ = E₀/η₀. The time-average power density is

Hence,

The units are consistent because (V/m)²/Ω = W/m². The result assumes a lossless free-space plane wave and uses peak, not rms, electric-field amplitude.

10. Illustrative application: inductive wireless charging pad

Background

An inductive charging pad transfers energy from a transmitter coil to a receiver coil without metallic contact. The transmitter is supplied with alternating current, producing a time-varying magnetic field. The receiver coil experiences changing magnetic flux and develops an induced emf.

Problem

A designer must transfer useful power across a small air gap while limiting heating, electromagnetic interference, and unsafe exposure. A static magnetic-field model is inadequate because the operating principle depends on time-varying flux.

Analysis

The transmitter current produces magnetic flux B. Faraday’s law predicts receiver-coil emf:

where N is the number of receiver turns. The negative sign establishes the polarity required by Lenz’s law. The coil geometry, alignment, frequency, and magnetic materials determine the mutual inductance and coupling coefficient. Ferrite material may guide flux and reduce leakage, but its permeability and losses are frequency-dependent.

The displacement-current term is also relevant in the electric field between conductors and across parasitic capacitances. The Poynting vector provides a field-based view of how power moves through the surrounding space and materials, rather than assuming that energy travels only inside the copper wires.

Approach and solution

A practical design process is to:

1. specify operating frequency, power, coil dimensions, and separation;
2. use a coupled-circuit approximation for preliminary calculations;
3. solve Maxwell’s equations numerically when coil geometry and shielding are complex;
4. evaluate conductor loss, core loss, dielectric loss, and thermal rise;
5. apply boundary conditions at copper, ferrite, air, and enclosure interfaces;
6. measure efficiency, stray field, temperature, and electromagnetic compatibility;
7. include control circuitry that adjusts frequency or excitation to maintain safe operation.

Outcome

A successful design delivers the required power with acceptable efficiency and temperature while satisfying applicable electromagnetic compatibility and product-safety requirements. Misalignment usually reduces mutual coupling, increases reactive circulating current, and can increase losses.

Learning points

The application demonstrates that Faraday’s law explains induced voltage, constitutive relations describe material response, boundary conditions govern field transitions, and the Poynting vector describes energy transport. It also illustrates why ideal formulas must be combined with numerical modelling, measurement, thermal analysis, and safety engineering.

11. Limitations, best practices, and contemporary developments

The simplest field formulas rely on idealizations: point charges, infinite conductors, homogeneous materials, linear response, negligible edge effects, or steady state. These assumptions should be stated before applying a formula. At high frequencies, material dispersion, radiation, skin effect, dielectric loss, and wave propagation may be important. At microscopic scales, classical macroscopic fields may require modification through quantum or statistical models.

Best practice includes choosing the coordinate system according to symmetry, checking units, identifying free and bound charge, distinguishing B from H and E from D, applying the correct boundary conditions, and verifying solutions with limiting cases. Numerical electromagnetic tools based on finite-element, finite-difference, or method-of-moments techniques are valuable for complex geometries, but their meshes, material data, boundary conditions, and convergence must be checked rather than accepted uncritically.

Modern applications include computational antenna design, wireless power, photonic devices, metamaterials, electric vehicles, magnetic-resonance systems, and electromagnetic compatibility analysis. Machine-learning tools can assist surrogate modelling or optimization, but they do not replace Maxwell-based validation, physical constraints, uncertainty analysis, or safety assessment. Engineers remain responsible for model assumptions and regulatory compliance.

Chapter Summary

Electric and magnetic fields provide a unified description of forces, energy storage, induction, and power transfer. Coulomb’s law and Gauss’s law govern electrostatic fields; polarization distinguishes free charge from bound charge in dielectrics. The Biot–Savart law and Ampère’s law describe magnetostatic fields, while Faraday’s law describes induced electric fields. Maxwell’s displacement-current correction restores charge continuity and completes the four Maxwell equations.

Boundary conditions specify changes in normal and tangential field components at interfaces. Constitutive relations connect fields to material properties. The Poynting vector, S = E × H, describes electromagnetic power flow, and Poynting’s theorem expresses energy conservation. These concepts are directly applied in transformers, motors, antennas, transmission systems, wireless chargers, and electromagnetic compatibility studies. Reliable engineering analysis requires appropriate assumptions, dimensional checks, material data, numerical verification, measurement, and attention to safety and regulation.

Key Terms

Electric field: Force per unit positive test charge.

Electric flux density: Field quantity D related to free charge through Gauss’s law.

Magnetic flux density: Vector B measured in teslas.

Magnetic field intensity: Vector H measured in amperes per metre.

Permittivity: Material property relating D to E.

Permeability: Material property relating B to H.

Polarization: Electric dipole moment per unit volume.

Free charge: Charge supplied or controlled by external sources.

Bound charge: Charge associated with polarization of matter.

Gauss’s law: Law relating electric flux through a closed surface to enclosed free charge.

Ampère–Maxwell law: Law relating magnetic circulation to conduction and displacement current.

Faraday’s law: Law stating that changing magnetic flux produces emf.

Lenz’s law: Rule that induced effects oppose the change producing them.

Displacement current: Time derivative of electric flux density.

Boundary condition: Field relation required at an interface between media.

Poynting vector: Vector E × H representing electromagnetic power-flow density.

Electromagnetic energy density: Energy stored per unit volume in electric and magnetic fields.

Constitutive relation: Material equation such as D = εE.

Review Questions

Short Answer Questions

1. What is the physical meaning and SI unit of electric field intensity?
2. Distinguish free charge from bound charge in a dielectric.
3. Why is the displacement-current term required in Ampère’s law?
4. State the boundary condition for the normal component of magnetic flux density.
5. What physical information is conveyed by the Poynting vector?

Long Answer Questions

1. Derive the electric field outside and inside a uniformly charged solid sphere using Gauss’s law.
2. Explain polarization, electric susceptibility, permittivity, and bound charge density.
3. State Maxwell’s four equations in integral and differential forms and explain their physical meanings.
4. Derive the electromagnetic boundary conditions using a Gaussian pillbox and a small rectangular loop.
5. Explain Poynting’s theorem and discuss the meaning of each term in an electrically conducting medium.

Analytical / Application Questions

1. A parallel-plate capacitor is being charged by a time-varying current. Explain how the conduction-current and displacement-current descriptions remain consistent, and identify the relevant Maxwell equation.
2. A wireless charging system becomes significantly less efficient when the receiver coil is laterally displaced. Use Faraday’s law, mutual coupling, losses, and Poynting-vector concepts to explain why.
3. At an interface with no free surface charge, medium 1 has ε₁ = 2ε₀ and medium 2 has ε₂ = 8ε₀. If the normal electric field in medium 1 is 400 V/m, calculate the normal electric field in medium 2 and state the assumption used.

Multiple Choice Questions with Answers

Question 1

Which Maxwell equation states that the divergence of electric flux density equals free-charge density?

A. ∇ · 𝐃 = ρ_f

B. ∇ · 𝐁 = 0

C. ∇ × 𝐄 = −∂𝐁/∂t

D. ∇ × 𝐇 = 𝐉 + ∂𝐃/∂t

Correct Answer: A. ∇ · 𝐃 = ρ_f

Question 2

For a linear isotropic dielectric, which constitutive relation is correct?

A. 𝐁 = ε𝐄

B. 𝐃 = ε𝐄

C. 𝐄 = μ𝐇

D. 𝐏 = μ𝐇

Correct Answer: B. 𝐃 = ε𝐄

Question 3

The displacement-current density is represented by which quantity?

A. ∇ · 𝐁

B. ∇ × 𝐄

C. ∂𝐃/∂t

D. ρ_f𝐄

Correct Answer: C. ∂𝐃/∂t

Question 4

A point charge in free space produces an electric-field magnitude that varies with distance as:

A. r

B. 1/r

C. 1/r²

D. 1/r³

Correct Answer: C. 1/r²

Question 5

What is the SI unit of the Poynting vector?

A. J

B. W/m²

C. C/m²

D. T

Correct Answer: B. W/m²

Question 6

At an interface with no free surface charge, which condition is correct?

A. The normal component of 𝐄 is always continuous.

B. The normal component of 𝐃 is continuous.

C. The tangential component of 𝐃 is continuous.

D. The normal component of 𝐇 is always zero.

Correct Answer: B. The normal component of 𝐃 is continuous.

Question 7

A plane wave in free space has peak electric-field magnitude 120 V/m. Using η₀ = 377 Ω, its average power density is approximately:

A. 0.159 W/m²

B. 19.1 W/m²

C. 45.3 W/m²

D. 377 W/m²

Correct Answer: B. 19.1 W/m²

Question 8

Which law explains the polarity of an induced emf opposing the change in magnetic flux?

A. Gauss’s electric law

B. Coulomb’s law

C. Lenz’s law

D. Gauss’s magnetic law

Correct Answer: C. Lenz’s law

Question 9

Why is ∇ · 𝐁 = 0 physically significant?

A. Electric charge is conserved.

B. Magnetic field has no isolated sources or sinks.

C. Electric fields are always static.

D. Conductivity is zero in free space.

Correct Answer: B. Magnetic field has no isolated sources or sinks.

Question 10

For a conductor in electrostatic equilibrium, the electric field inside the conductor is:

A. Uniform and nonzero

B. Radially increasing

C. Zero

D. Equal to the magnetic field

Correct Answer: C. Zero