Unit 5 Electromagnetism: From Fields to Light

Maxwell’s equations

Four consistent laws that describe how electric fields and magnetic fields are sourced by charge/current and how they change in time, working together everywhere in space and time.

Field (physical viewpoint)

A real physical entity created by charges/currents that can store energy and can generate other fields when it varies in time (even in empty space).

Integral form (of a field law)

A version of a Maxwell equation that relates what happens inside a region to the field on the boundary via a surface flux integral or a loop circulation integral.

Differential form (of a field law)

A point-by-point version of a Maxwell equation using divergence or curl to describe local sources or circulation of a field.

Electric flux (ΦE)

Measure of electric field passing through a surface: ΦE = ∫ E⃗ · dA⃗.

Magnetic flux (ΦB)

Measure of magnetic field passing through a surface: ΦB = ∫ B⃗ · dA⃗.

Permittivity of free space (ε0)

A constant that sets the strength of electric field effects in vacuum; appears in Gauss’s law and the EM wave speed c = 1/√(μ0ε0).

Permeability of free space (μ0)

A constant that sets the strength of magnetic field effects in vacuum; appears in Ampère–Maxwell law and the EM wave speed c = 1/√(μ0ε0).

Gauss’s law for electricity

Electric flux through a closed surface equals enclosed charge divided by ε0: ∮ E⃗ · dA⃗ = Qenc/ε0; charges are sources/sinks of E⃗.

Enclosed charge (Qenc)

The net charge inside a chosen closed Gaussian surface; it determines the net electric flux through that surface in Gauss’s law.

Charge density (ρ)

Charge per unit volume; appears in the differential Gauss’s law ∇·E⃗ = ρ/ε0.

Divergence (∇·)

A measure of net “outflow” of a vector field from a point; in E&M it indicates where a field has sources/sinks (e.g., ∇·E⃗ relates to charge).

Gauss’s law for magnetism

Net magnetic flux through any closed surface is zero: ∮ B⃗ · dA⃗ = 0; magnetic field lines have no beginning or end in classical E&M.

Magnetic monopole (classical context)

A hypothetical isolated magnetic “charge”; Gauss’s law for magnetism (∮ B⃗ · dA⃗ = 0) encodes that monopoles are not observed in classical electromagnetism.

Faraday’s law of induction

A changing magnetic flux through a loop induces emf/circulating electric field: ℰ = −dΦB/dt and ∮ E⃗ · dℓ⃗ = −dΦB/dt.

Electromotive force (emf, ℰ)

The loop integral of the electric field around a closed path (units of volts); in Faraday’s law it equals −dΦB/dt.

Lenz’s law (minus sign)

The induced emf/current acts to oppose the change in magnetic flux that produced it, captured by the negative sign in ℰ = −dΦB/dt.

Non-conservative electric field

An electric field with nonzero loop integral (∮ E⃗ · dℓ⃗ ≠ 0), which can occur when magnetic flux changes; a single global electric potential cannot always be defined.

Curl (∇×)

A measure of local “circulation density” (swirl) of a vector field; e.g., ∇×E⃗ = −∂B⃗/∂t and ∇×B⃗ relates to current and changing E⃗.

Ampère–Maxwell law

Magnetic circulation around a loop is produced by conduction current and changing electric flux: ∮ B⃗ · dℓ⃗ = μ0 Ienc + μ0ε0 dΦE/dt.

Displacement current (Id)

Effective current term from changing electric flux: Id = ε0 dΦE/dt; ensures Ampère’s law works consistently in time-varying situations (e.g., charging capacitors).

Electromagnetic wave

A self-propagating traveling disturbance in which E⃗ and B⃗ oscillate, perpendicular to each other and to the direction of propagation; does not require a material medium.

Speed of light in vacuum (c)

The wave speed predicted by Maxwell’s equations in vacuum: c = 1/√(μ0ε0); independent of frequency in vacuum.

Field magnitude relation in a vacuum plane wave (E = cB)

In a plane electromagnetic wave in vacuum, the magnitudes satisfy E = cB and are in phase (rise and fall together).

Poynting vector (S⃗)

Vector giving direction and rate of electromagnetic energy flow: S⃗ = (1/μ0) E⃗ × B⃗; points in the propagation direction for a plane wave.