Vector Fields, Helmholtz Decomposition, and Charge Densities (Vocabulary)

Vocabulary flashcards covering divergence, curl, gradient, Helmholtz decomposition, charge densities, potentials, and related Maxwell concepts, based on the lecture notes.

Divergence (∇·F)

A scalar measure of a vector field’s net outflow or inflow at a point; indicates sources or sinks and relates to charge density via Gauss’s law.

Curl (∇×F)

A vector measure of a field’s local rotation or circulation; nonzero curl indicates rotation around a point and relates to changing magnetic fields.

Gradient (∇f)

The vector of partial derivatives of a scalar field f; points in the direction of greatest increase; used to express potential fields (e.g., E = -∇V in electrostatics).

Helmholtz decomposition

Any smooth vector field can be written as F = -∇φ + ∇×A, separating a curl-free part and a divergence-free part.

Conservative field

A vector field with zero curl; line integrals between two points are path-independent; F = -∇φ.

Divergence-free field

A field with zero divergence; often expressible as a curl, e.g., B = ∇×A, indicating no net sources or sinks.

Electric field (E)

Field due to charges; in electrostatics, curl is zero (∇×E = 0) and E = -∇V.

Magnetic field (B)

Field related to moving charges/currents; has ∇·B = 0 and can be written B = ∇×A.

Vector potential (A)

A vector field whose curl gives the magnetic field: B = ∇×A; helps enforce ∇·B = 0.

Volumetric charge density (ρ_v)

Charge per unit volume (C/m^3); total charge in a region is Q = ∫ ρ_v dV.

Surface charge density (ρ_s)

Charge per unit area on a surface (C/m^2); total charge on a surface S is Q = ∮ ρ_s dS.

Line charge density (ρ_l)

Charge per unit length along a line (C/m); total charge on a line is Q = ∫ ρ_l dl.

Total charge

Net charge in a region obtained by integrating the appropriate density over volume, surface, or line.

Current density (J)

Current per unit area crossing a surface; vector with units A/m^2; direction follows the flow; dI = J·dS.

Electric potential (V)

Scalar potential whose gradient gives the electric field: E = -∇V; describes the stored energy per unit charge.

Line integral of E

The integral ∫ E·dl between two points; equals -ΔV if E is conservative; shows voltage is path-independent in electrostatics.

Curl of a gradient is zero

∇×(∇f) = 0 for any scalar f; a fundamental vector calculus identity.

Divergence of a curl is zero

∇·(∇×A) = 0 for any vector A; another fundamental vector calculus identity.

Faraday’s law (time-varying B)

∇×E = -∂B/∂t; a changing magnetic field induces curl in the electric field.

Boundary effects / flux

Field behavior inside a region is influenced by what happens at the boundary; flux through the boundary (e.g., ∮ E·n dA) and boundary conditions affect the internal field.