Volume Charge Density Formula: ρ = q/V, Unit and Examples

The volume charge density formula is $\rho = \frac{q}{V}$ : divide the charge in a region by the volume of that region. The result says how many coulombs sit in each cubic metre, so the SI unit is the coulomb per cubic metre (C/m³).

You need this quantity whenever charge fills a solid, such as a charged insulating ball, a cloud of ions or the space charge near a diode junction. Charge on the outside of a conductor is a surface case, and charge along a thin wire is a line case, so each has its own density. The three versions are compared on the charge density formula page, part of the electricity and electrostatics formulas collection.

Quick facts

Formula
$\rho = \frac{q}{V}$
SI unit
coulomb per cubic metre (C/m³)
Dimensional formula
[M⁰ L⁻³ T¹ I¹]
Type
Scalar, can be positive or negative
Non-uniform case
$q = \int \rho\, dV$
Symbol clash
ρ is also used for mass density and resistivity

Meaning of each term in the volume charge density formula

Terms in ρ = q / V
Symbol	Meaning	SI unit
$\rho$	Charge per unit volume (Greek rho)	C/m³
$q$	Net charge inside the chosen volume	C
$V$	Volume occupied by that charge	m³

The sign of ρ follows the sign of q. A ball carrying extra electrons has negative ρ. If the charge is spread evenly every cubic metre holds an equal share and plain division is enough. If it is not even, ρ changes from point to point and you add up thin shells: $q = \int \rho\, dV$ . For a sphere where ρ depends only on radius, $dV = 4\pi r^2\, dr$ .

Working out ρ, q or V

Write down the shape and find its volume: sphere $\frac{4}{3}\pi R^3$ , cylinder $\pi R^2 L$ , cube $a^3$ .
Change lengths to metres before you cube them. A cube of side 2 cm is $(0.02)^3 = 8 \times 10^{-6}\ \text{m}^3$ , not 8 m³.
Change μC, nC or pC to coulombs.
Divide or multiply as needed and keep the sign of the charge.

The calculator on this page takes any two of ρ, q and V in SI units and gives the third, which is handy for checking powers of ten.

Volume charge density examples

Example 1: a cube of charged plastic

A plastic cube of side 2.0 cm has a uniform charge density of 5.0 × 10⁻⁴ C/m³. How much charge does it hold?

Volume: $V = (0.020)^3 = 8.0 \times 10^{-6}\ \text{m}^3$

$q = \rho V = (5.0 \times 10^{-4})(8.0 \times 10^{-6}) = 4.0 \times 10^{-9}\ \text{C} = 4.0\ \text{nC}$

Example 2: an insulating sphere (exam style)

A glass bead of radius 4.0 cm holds 1.2 μC, shared out evenly through its bulk. Find ρ.

Convert: R = 0.040 m, q = 1.2 × 10⁻⁶ C.

$V = \frac{4}{3}\pi (0.040)^3 = 2.68 \times 10^{-4}\ \text{m}^3$

$\rho = \frac{1.2 \times 10^{-6}}{2.68 \times 10^{-4}} = 4.5 \times 10^{-3}\ \text{C/m}^3$

Cubing a length in centimetres and forgetting to convert gives an answer off by a factor of one million.

Linking ρ to surface and line densities

Three ways charge can be spread
Spread over	Density	Unit	Typical body
A volume	$\rho = q/V$	C/m³	Charged insulating ball
A surface	$\sigma = q/A$	C/m²	Conducting sphere or plate
A line	$\lambda = q/L$	C/m	Thin charged rod or wire

A conductor in equilibrium has ρ = 0 inside, because all excess charge moves to its outer surface. That is why problems about metal spheres use the surface charge density formula, while thin rods use the linear charge density formula.

Where ρ appears in Gauss's law

Gauss's law says the flux leaving any closed surface is the charge inside divided by $\varepsilon_0$ . For a uniformly charged sphere of radius R, a Gaussian sphere of radius r inside it encloses $\rho \cdot \frac{4}{3}\pi r^3$ . That gives the field inside the ball:

$E = \frac{\rho r}{3\varepsilon_0} \quad (r \le R)$

So E rises in a straight line from nothing at the middle of the ball up to a peak on its skin. Beyond the skin it drops off with $1/r^2$ . More on fields from charge distributions is on the electric field formula page.

For dimensions, write coulomb as ampere times second, [I T], and divide by cubic length [L³]; ρ ends up with [M⁰ L⁻³ T¹ I¹] overall. Since 1 C = 1 A s, the unit can also be written A s/m³.

Errors to avoid

Confusing the three ρ. In this chapter ρ is charge density, not mass density (kg/m³) or resistivity (Ω m). Read the units in the question.
Using a conductor's volume. Excess charge on a metal sits on its surface; use σ, not ρ.
Hollow shapes. For a thick shell, the volume is $\frac{4}{3}\pi (R_2^3 - R_1^3)$ , not the volume of the outer sphere.
Dropping the sign. Negative charge gives negative ρ, which flips the direction of the field you work out.

Frequently asked questions

What is the volume charge density formula?

Use ρ = q / V: total coulombs in a body over the cubic metres it fills. When the charge is not evenly spread, the total is found with q = ∫ρ dV.

What is the SI unit of volume charge density?

The SI unit is the coulomb per cubic metre, C/m³. In base units that is A s/m³.

How are the dimensions of ρ worked out?

Take charge as current multiplied by time, [I T], then share it over a volume [L³]. That leaves [M⁰ L⁻³ T¹ I¹] for volume charge density.

What is the difference between volume, surface and linear charge density?

Volume density ρ is charge per cubic metre, surface density σ is charge per square metre and linear density λ is charge per metre. Choose the one that matches how the charge is spread.

Is volume charge density zero inside a conductor?

Yes, in electrostatic equilibrium. Any extra charge on a conductor moves to its outer surface, so the inside has no net charge.