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Formulas/physics/Electric Charges Fields/Linear, Surface, and Volume Charge Densities

Linear, Surface, and Volume Charge Densities

Charge densities for distributed charges. λ in C/m, σ in C/m², ρ in C/m³. Used to set up field integrals for continuous distributions.
Class 11Class 12
Derivation

Why charge densities

When charge is distributed continuously over a region rather than concentrated at discrete points, we need a density function to set up field integrals. Three cases arise depending on the geometry.

Linear charge density λ\lambda

For charge distributed along a curve (a wire, a ring):

λ=dqdl[C m−1]\lambda = \frac{dq}{dl} \quad [\text{C m}^{-1}]

Total charge on a segment from l1l_1 to l2l_2:

Q=∫l1l2λ dlQ = \int_{l_1}^{l_2} \lambda\, dl

For a uniform distribution: Q=λLQ = \lambda L, where LL is the total length.

Surface charge density σ\sigma

For charge distributed over a surface (a sheet, a spherical shell):

σ=dqdA[C m−2]\sigma = \frac{dq}{dA} \quad [\text{C m}^{-2}]

Total charge on a surface:

Q=∫Sσ dAQ = \int_S \sigma\, dA

For a uniform distribution: Q=σAQ = \sigma A.

Volume charge density ρ\rho

For charge distributed throughout a volume (a solid sphere, a cloud):

ρ=dqdV[C m−3]\rho = \frac{dq}{dV} \quad [\text{C m}^{-3}]

Total charge in a volume:

Q=∫Vρ dVQ = \int_V \rho\, dV

For a uniform distribution: Q=ρVQ = \rho V.

λ=dqdl,σ=dqdA,ρ=dqdV\boxed{\lambda = \frac{dq}{dl}, \quad \sigma = \frac{dq}{dA}, \quad \rho = \frac{dq}{dV}}

Use in field integrals

The superposition principle for continuous distributions:

E⃗(r⃗)=14πε0∫ρ(r⃗′)(r⃗−r⃗′)∣r⃗−r⃗′∣3 dV′\vec{E}(\vec{r}) = \frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\vec{r}')(\vec{r}-\vec{r}')}{|\vec{r}-\vec{r}'|^3}\, dV'

Replace ρ dV′\rho\, dV' with σ dA′\sigma\, dA' or λ dl′\lambda\, dl' for surface or line distributions respectively.

Note
These are not three separate types of object — a surface charge density $\sigma$ can be modelled as a volume density $\rho$ concentrated in a thin layer of thickness $\delta \to 0$ such that $\rho\,\delta = \sigma$. The density type is a modelling choice based on geometry.