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Charge Density Formula

Electricity and magnetism formulasScalarClass 12Updated 1 Nov 2026

ρ=qV\rho = \frac{q}{V}
SI unit
C/m³
Dimensions
[M⁰ L⁻³ T¹ I¹]
where
  • ρ volume charge density C/m³
  • q charge enclosed C
  • V volume occupied by the charge m³

The charge density formula in its volume form is ρ=qV\rho = \frac{q}{V}: the charge in a region divided by the volume of that region, in coulombs per cubic metre. When charge lies on a surface or along a line instead, you switch to σ=qA\sigma = \frac{q}{A} or λ=qL\lambda = \frac{q}{L}.

Which one a question wants depends on the shape. A solid ball of charge needs ρ, a metal plate needs σ, and a long thin wire needs λ. This page sets out all three side by side, works an example of each, and shows how they feed into the field formulas you use in Class 12 and AP Physics 2.

Quick facts

  • Volume

    ρ=qV\rho = \frac{q}{V}, unit C/m³

  • Surface

    σ=qA\sigma = \frac{q}{A}, unit C/m²

  • Line

    λ=qL\lambda = \frac{q}{L}, unit C/m

  • Dimensional formula of ρ

    [M⁰ L⁻³ T¹ I¹]

  • Type

    Scalar, positive or negative

  • Carrier form

    ρ=ne\rho = ne for n carriers per m³

Choosing the right charge density formula

Three densities for three shapes
KindSymbol and formulaSI unitTypical object
Volumeρ=q/V\rho = q/VC/m³Uniformly charged sphere or cylinder
Surfaceσ=q/A\sigma = q/AC/m²Conducting plate, spherical shell
Lineλ=q/L\lambda = q/LC/mThin rod, wire, ring

Ask yourself how many dimensions the charge is spread over. Three dimensions means ρ, two means σ, one means λ. Each is an average if the charge is uneven; the point value uses a small element, such as ρ=dqdV\rho = \frac{dq}{dV}.

Worked examples for each type

Example 1: a charged ball

A solid insulating sphere of radius 5.0 cm holds 2.0 μC spread through its volume. Find ρ.

V = (4/3)π(0.050)³ = 5.24 × 10⁻⁴ m³.

ρ=2.0×10−65.24×10−4≈3.8×10−3 C/m3\rho = \frac{2.0\times10^{-6}}{5.24\times10^{-4}} \approx 3.8\times10^{-3}\ \text{C/m}^3

Example 2: a thin rod

A 2.0 m rod carries 6.0 nC evenly along its length. Find λ.

λ=6.0×10−92.0=3.0×10−9 C/m\lambda = \frac{6.0\times10^{-9}}{2.0} = 3.0\times10^{-9}\ \text{C/m}

Each metre of the rod carries 3 nC.

Example 3: a disc

A disc of radius 0.10 m holds 3.14 nC on one face. Find σ.

A = π(0.10)² = 0.0314 m².

σ=3.14×10−90.0314=1.0×10−7 C/m2\sigma = \frac{3.14\times10^{-9}}{0.0314} = 1.0\times10^{-7}\ \text{C/m}^2

Volume formulas and conversions

Volumes and unit changes
ItemValue
Sphere of radius R43πR3\frac{4}{3}\pi R^3
Cylinder of radius r, length Lπr2L\pi r^2L
One cubic centimetrea millionth of a cubic metre (10⁻⁶ m³)
One litrea thousandth of a cubic metre (10⁻³ m³)
1 μC/cm³1 C/m³

The last row surprises students: the 10⁻⁶ in the microcoulomb cancels the 10⁻⁶ in the cubic centimetre. Always write the powers of ten out before cancelling.

Charge density and the field inside a charged sphere

Inside a uniformly charged ball, only the charge within radius r counts. That charge is ρ⋅43πr3\rho\cdot\frac{4}{3}\pi r^3, and Gauss's law gives

E=ρr3ε0(r<R)E = \frac{\rho r}{3\varepsilon_0} \quad (r < R)

So the field grows in a straight line from zero at the centre to its largest value at the surface, then falls as 1/r² outside. Field results for sheets and lines use the surface and linear densities, and are listed on the electric field page.

Charge density of carriers in a metal

In current electricity, the same idea describes the mobile electrons. If a metal has n free electrons per cubic metre, their charge density is nene. For copper, n ≈ 8.5 × 10²⁸ m⁻³, so the mobile charge is about 1.4 × 10¹⁰ C/m³, cancelled exactly by the positive ions. That huge reservoir is why a small drift velocity is enough to carry everyday currents, through I=neAvdI = neAv_d.

Do not confuse ρ for charge density with ρ for resistivity or for mass density. The unit tells you which one is meant.

Avoiding common errors

  • Using a surface area when the charge fills a volume, or the other way round.
  • Cubing the diameter instead of the radius.
  • Dropping the negative sign on electron charge densities.
  • Mixing up the three Greek letters: λ for line, σ for surface, ρ for volume.

The detailed page for the volume form is volume charge density. The electricity formulas hub lists every page in this chapter.

Frequently asked questions

What is the formula for charge density?

For a volume, ρ = q / V. For a surface, σ = q / A. For a line, λ = q / L.

Which units go with each kind of charge density?

Volume charge density is in C/m³, surface charge density in C/m² and linear charge density in C/m.

Is charge density a scalar?

Yes. It has no direction, only a size and a sign.

What is the dimensional formula of volume charge density?

Charge is [I T] and volume is [L³], so ρ works out to [M⁰ L⁻³ T¹ I¹].

How is charge density different from charge?

Charge is the total amount in coulombs. Charge density says how that amount is spread, per metre, square metre or cubic metre.