# Charge density
**Charge density** is the amount of electric charge per unit length, area, or volume of a region of space. It generalizes the idea of a point charge to distributions of charge spread over a line, a surface, or a volume, and it is the quantity from which electric fields and potentials of extended charged objects are calculated. The three standard forms are the linear charge density λ (charge per unit length, in coulombs per meter), the surface charge density σ (charge per unit area, in coulombs per square meter), and the volume charge density ρ (charge per unit volume, in coulombs per cubic meter)[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions). The SI recognizes the same units: electric charge density in C/m³ and surface charge density in C/m²[2](https://www.nist.gov/pml/special-publication-330/sp-330-section-2).
| Key fact | Detail |
|---|---|
| Linear density | λ = charge per unit length; SI unit C/m[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions) |
| Surface density | σ = charge per unit area; SI unit C/m²[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions) |
| Volume density | ρ = charge per unit volume; SI unit C/m³[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions) • [2](https://www.nist.gov/pml/special-publication-330/sp-330-section-2) |
| Differential charge | dq = λ dl for a line, σ dA for a surface, ρ dV for a volume[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions) |
| Conductor surface field | Just above a conductor in electrostatic equilibrium, E = σ/ε₀, directed perpendicular to the surface[3](https://openstax.org/books/university-physics-volume-2/pages/6-4-conductors-in-electrostatic-equilibrium) |
| Curvature effect | On a conductor, σ and E are greater at regions with smaller radii of curvature[4](https://pressbooks.online.ucf.edu/uphysicstjb/chapter/equipotential-surfaces-and-conductors/) |
| Conservation | Charge density obeys the continuity equation ∇·J = −∂ρ/∂t, the local form of charge conservation[5](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/6fa0dd1b76ca434baefa294742476195_MIT6_013S09_back.pdf) |
## The three densities
Each density is defined as a limit: the charge in a small portion of the object divided by the size of that portion. For a one-dimensional line, λ = dq/dx; for a two-dimensional surface, σ = dq/dA, where dA is a small area element; for a three-dimensional volume, ρ = dq/dV, where dV is a small volume element[6](https://lipa.physics.oregonstate.edu/sec_charge-density.html). Like mass density, charge density describes how charge is distributed throughout a line, a surface, or a volume[6](https://lipa.physics.oregonstate.edu/sec_charge-density.html).
The densities need not be constant. A wire of length L whose density changes from one end to the other does not carry a total charge q = λL; the total charge must be found by integrating the varying density[6](https://lipa.physics.oregonstate.edu/sec_charge-density.html). Texts therefore often write σ(r) and λ(r) as functions of position[7](https://books.physics.oregonstate.edu/GVC/densities.html).
## Continuous distributions as an approximation
Because electric charge is quantized, no charge distribution is truly continuous. The continuous description works because in most practical cases the total charge involves such a huge number of discrete charges that the discrete nature can be ignored, in the same way a bucket of water is treated as a continuous fluid rather than a collection of H₂O molecules[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions).
Once a distribution is treated as continuous, the electric field at a point P follows by dividing the charge into infinitesimal pieces, treating each as a point charge, and integrating. The point-charge sum becomes an integral with q_i replaced by dq = λ dl, σ dA, or ρ dV for line, surface, and volume charges respectively[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions):
\[ \vec{E}(P) = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r^2}\, \hat{r} \]
where r is the distance from the charge element to P. The integrals assume the principle of superposition, and the practical difficulty is usually writing dl, dA, or dV correctly in terms of r and expressing the density function, which may be constant or position-dependent[1](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions).
## Idealized and singular densities
Line and surface densities are idealizations of charge concentrated in a very thin region around a one-dimensional path or two-dimensional surface[7](https://books.physics.oregonstate.edu/GVC/densities.html). The [Dirac delta function](https://www.edgepedia.org/dirac-delta-function) connects these idealizations to volume densities: a surface density σ(x,y) concentrated at z = 0 can be written as the volume density ρ(x,y,z) = σ(x,y) δ(z), and integrating this over a solid region gives the same total charge as integrating σ over the surface[8](https://books.physics.oregonstate.edu/GSF/deltadensity.html). A point charge is the limit of an infinite charge density occupying zero volume, with the net charge q finite[9](https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/e9209c908e2d840ed58fb6cf2f925279_01.pdf).
## Charge density on conductors
In electrostatic equilibrium the electric field inside a conductor is zero, and any excess charge must lie on its surface[3](https://openstax.org/books/university-physics-volume-2/pages/6-4-conductors-in-electrostatic-equilibrium). Just above the surface, the surface charge density and the field magnitude are related by E = σ/ε₀, derived with an infinitesimal cylindrical [Gaussian surface](https://www.edgepedia.org/gaussian-surface) straddling the surface; the field is always perpendicular to the surface, since any parallel component would move the free charges[3](https://openstax.org/books/university-physics-volume-2/pages/6-4-conductors-in-electrostatic-equilibrium).
The distribution over the surface is generally not uniform. On a conductor with varying curvature, σ and E are greater at regions with smaller radii of curvature[4](https://pressbooks.online.ucf.edu/uphysicstjb/chapter/equipotential-surfaces-and-conductors/). A lightning rod exploits this: charge accumulating at the sharp end produces a very large field there, and when the field reaches approximately 3.0 × 10⁶ N/C, the dielectric strength of air, air molecules ionize and charge flows through the rod to Earth[4](https://pressbooks.online.ucf.edu/uphysicstjb/chapter/equipotential-surfaces-and-conductors/).
## Time dependence and conservation
When charge moves, the density ρ(x, y, z, t) and the current density J are linked by the continuity equation,
\[ \nabla \cdot \vec{J} = -\frac{\partial \rho}{\partial t} \]
which states that charge flowing out of a volume decreases the charge within it; this is the local form of the law of electric charge conservation[5](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/6fa0dd1b76ca434baefa294742476195_MIT6_013S09_back.pdf) • [10](https://web2.ph.utexas.edu/~vadim/Classes/2017f/continuity.pdf). The equation follows from [Maxwell's equations](https://www.edgepedia.org/maxwells-equations), and some treatments take charge conservation as the fundamental axiom from which the equations can instead be derived[11](https://arxiv.org/abs/0812.4785).
## References
1. [5.5 Calculating Electric Fields of Charge Distributions, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/5-5-calculating-electric-fields-of-charge-distributions)
2. [SP 330 - Section 2, The International System of Units, NIST](https://www.nist.gov/pml/special-publication-330/sp-330-section-2)
3. [6.4 Conductors in Electrostatic Equilibrium, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/6-4-conductors-in-electrostatic-equilibrium)
4. [7.5 Equipotential Surfaces and Conductors, University Physics Volume 2](https://pressbooks.online.ucf.edu/uphysicstjb/chapter/equipotential-surfaces-and-conductors/)
5. [Appendix D: Basic Equations for Electromagnetics and Applications, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-013-electromagnetics-and-applications-spring-2009/6fa0dd1b76ca434baefa294742476195_MIT6_013S09_back.pdf)
6. [Charge Density, Paradigms in Physics, Oregon State University](https://lipa.physics.oregonstate.edu/sec_charge-density.html)
7. [Densities, Paradigms in Physics, Oregon State University](https://books.physics.oregonstate.edu/GVC/densities.html)
8. [The Dirac Delta Function and Densities, Paradigms in Physics, Oregon State University](https://books.physics.oregonstate.edu/GSF/deltadensity.html)
9. [Electromagnetic Fields, Forces, and Motion, Chapter 1, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-641-electromagnetic-fields-forces-and-motion-spring-2005/e9209c908e2d840ed58fb6cf2f925279_01.pdf)
10. [Electric Charge Conservation and the Continuity Equation, University of Texas at Austin](https://web2.ph.utexas.edu/~vadim/Classes/2017f/continuity.pdf)
11. [Can Maxwell's equations be obtained from the continuity equation? arXiv](https://arxiv.org/abs/0812.4785)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electromagnetic quantities and history › Electromagnetic quantities › Electric charge and current quantities*
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