1.3 Charge Distributions

Discrete vs. Continuous Charge

Real charge distributions are made of discrete point charges, but for macroscopic objects we model charge as continuous, spread smoothly over a region. This lets us use density functions — linear (λ), surface (σ), or volume (ρ) charge density — to describe how charge is distributed, and integrate over the object instead of summing point charges.

Charge Density Definitions

Linear charge density λ = dq/dL (C/m) for charge along a wire or rod. Surface charge density σ = dq/dA (C/m²) for charge spread over a surface. Volume charge density ρ = dq/dV (C/m³) for charge filling a 3D region. Total charge is found by integrating: Q = ∫λ dL, ∫σ dA, or ∫ρ dV over the object.

Setting Up Charge Elements

To find fields or potentials from continuous distributions, break the object into infinitesimal charge elements dq, express dq using the appropriate density (e.g., dq = λdL), then integrate the contribution of each dq over the whole shape. This is the foundation for calculating E fields from rods, rings, disks, and shells using Coulomb's Law integrals.

Key terms

Linear charge density (λ)

Charge per unit length: λ = dq/dL, units C/m. Used for wires and rods.

Surface charge density (σ)

Charge per unit area: σ = dq/dA, units C/m². Used for charged surfaces or sheets.

Volume charge density (ρ)

Charge per unit volume: ρ = dq/dV, units C/m³. Used for charge filling 3D regions.

dq

An infinitesimal element of charge, expressed as λdL, σdA, or ρdV depending on geometry, used in integration.

Why use continuous charge distributions?

They let us model macroscopic charged objects with integral calculus instead of summing countless discrete point charges.

Try an AP-style question

A rod from x = 0 to x = 3 m has linear charge density λ = 2x (C/m). What is its total charge?

1. 9 C
2. 3 C
3. 18 C
4. 6 C