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: $\Phi_E = \int \mathbf{E} \cdot d\textbf{A}$.

Magnetic flux (ΦB)

Measure of magnetic field passing through a surface: $\Phi_B = \oint \mathbf{B} \cdot d\textbf{A}$.

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 $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.

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 $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.

Gauss’s law for electricity

Electric flux through a closed surface equals enclosed charge divided by $\epsilon_0$: \noint \textbf{E} \bullet d\textbf{A} = \frac{Q_{enc}}{\epsilon_0}; charges are sources/sinks of $\textbf{E}$.

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 $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$.

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., $\nabla \cdot \mathbf{E}$ relates to charge).

Gauss’s law for magnetism

Net magnetic flux through any closed surface is zero: \noint \textbf{B} \bullet d\textbf{A} = 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 $\oint \mathbf{B} \cdot d\textbf{A} = 0$ 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: \text{ℰ} = -\frac{dΦ_B}{dt} and \noint \textbf{E} \bullet d\textbf{ℓ} = -\frac{dΦ_B}{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 \text{ℰ} = -\frac{dΦ_B}{dt}.

Non-conservative electric field

An electric field with nonzero loop integral (\noint \textbf{E} \bullet d\textbf{ℓ} \neq 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., $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ and $\nabla \times \mathbf{B}$ relates to current and changing $\mathbf{E}$.

Ampère–Maxwell law

Magnetic circulation around a loop is produced by conduction current and changing electric flux: \noint \textbf{B} \bullet d\textbf{ℓ} = bc_0 I_{enc} + bc_0\epsilon_0 \frac{dΦ_E}{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 = \frac{1}{\sqrt{bc_0\epsilon_0}}; 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: \textbf{S} = \frac{1}{bc_0} \textbf{E} \times \textbf{B}; points in the propagation direction for a plane wave.