Charge density

Charge density is charge per unit (length, area, or volume) — the way to describe distributed charge as a smooth field instead of a sum over discrete point charges.

Three flavors, depending on the dimension of the support:

Volume charge density ${\rho }_v$ (C/m³):

$Q={\int }_V{\rho }_{v}dv.$

Volume charge density ${\rho }_v$ assigned to each infinitesimal element $d{v}^{\prime }$ of a region. Total charge is the integral; the differential element ${\rho }_{v}d{v}^{\prime }$ acts like a “point charge” in Coulomb-law superposition.

Used when charge fills a 3D region — e.g., the interior of a charged semiconductor, the bulk of a plasma.

Surface charge density ${\rho }_s$ (C/m²):

$Q={\int }_S{\rho }_{s}ds.$

Used when charge accumulates on a 2D surface — the surface of a conductor, a plate of a parallel-plate capacitor, an interface where free charge piles up.

Line charge density ${\rho }_l$ (C/m):

$Q={\int }_C{\rho }_{l}dl.$

Used when charge is confined to a 1D filament — an idealized infinite wire, a thin charged ring.

Lower-dimensional densities are idealizations: physical charges always have finite extent in 3D, but treating them as confined to a surface or line often simplifies the math without losing essential physics.

Why charge density and not just total $Q$

Two reasons:

1. Where the charge sits matters as much as how much there is. Coulomb’s law for a single point charge says $\mathbf{E}$ depends on the Distance vector from the charge — you need to know the position of every bit of charge to compute the field. The charge density $\rho ({\mathbf{R}}^{\prime })$ at source point ${\mathbf{R}}^{\prime }$ is the integrand: $\mathbf{E}(\mathbf{R})=\frac{1}{4\pi ϵ}\int \frac{{\rho }_v({\mathbf{R}}^{\prime })(\mathbf{R}-{\mathbf{R}}^{\prime })}{|\mathbf{R}-{\mathbf{R}}^{\prime }{|}^3}d{v}^{\prime }.$

2. Maxwell’s equations are local. They tie field derivatives to local source density, not total enclosed charge. $\nabla \cdot \mathbf{D}={\rho }_v$ holds pointwise. The integral statements (Gauss’s law) are downstream of the local one.

Going between densities

Surface charge as a limit of volume charge: if a thin slab of thickness $\Delta$ has uniform volume density ${\rho }_v$, taking $\Delta \to 0$ with ${\rho }_s={\rho }_v\Delta$ held fixed gives a surface density ${\rho }_s$. Mathematically, ${\rho }_v$ becomes a delta function ${\rho }_s\delta (z-z_0)$ across the surface.

Similar for line charge: a thin tube of cross-section $A$ with volume density ${\rho }_v$ has line density ${\rho }_l={\rho }_{v}A$.

Current density

The moving analog of charge density is Current density $\mathbf{J}={\rho }_v\mathbf{u}$ (A/m²): the charge flux per unit area for a volume charge moving with velocity $\mathbf{u}$. It feeds the right-hand side of Ampère’s law in magnetostatics and is the magnetic counterpart to the role ${\rho }_v$ plays for the electric field.

The static charge density ${\rho }_v$ and dynamic current density $\mathbf{J}$ are linked by the Charge continuity equation $\nabla \cdot \mathbf{J}=-{\partial }_t{\rho }_v$ — charge conservation in differential form.