Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume. It is measured in coulombs per metre (C/m), square metre (C/m²), or cubic metre (C/m³), respectively. Since there are positive as well as negative charges, the charge density can take on negative values. Like any density it can depend on position. It should not be confused with the charge carrier density. As related to chemistry, it can refer to the charge distribution over the volume of a particle, molecule, or atom. Therefore, a lithium cation will carry a higher charge density than a sodium cation due to its smaller ionic radius.

Continuous charges
The integral of the charge density $$\alpha_q(\mathbf r)$$, $$\sigma_q(\mathbf r)$$, $$\rho_q(\mathbf r)$$ over a line $$l$$, surface $$S$$, or volume $$V$$, is equal to the total charge $$Q$$ of that region, defined to be:


 * $$Q=\int\limits_L \alpha_q(\mathbf r) dl$$,
 * $$Q=\int\limits_S \sigma_q(\mathbf r) dS$$,
 * $$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V.$$

This relation defines the charge density mathematically. Note that the symbols used to denote the various dimensions of charge density vary between fields of studies. Other commonly used notations are $$\lambda$$, $$\sigma$$, $$\rho$$; or $$\rho_l$$, $$\rho_s$$, $$\rho_v$$ for (C/m), (C/m²), (C/m³) and respectively.

Homogeneous charge density
For the special case of a homogeneous charge density, that is one that is independent of position, equal to $$\rho_{q,0}$$ the equation simplifies to:
 * $$Q=V\cdot \rho_{q,0}$$

The proof of this is simple. Start with the definition of the charge of any volume:
 * $$Q=\int\limits_V \rho_q(\mathbf r) \,\mathrm{d}V$$

Then, by definition of homogeneity, $$\rho_q(\mathbf r)$$ is a constant that we will denote $$\rho_{q,0}$$ to differentiate between the constant and non-constant forms, and thus by the properties of an integral can be pulled outside of the integral resulting in:
 * $$Q=\rho_{q,0} \int\limits_V \,\mathrm{d}V$$

Again, by the properties of integrals:
 * $$\int\limits_V \,\mathrm{d}V = V$$

Therefore by substitution:
 * $$\rho_{q,0} \int\limits_V \,\mathrm{d}V = V\cdot \rho_{q,0}$$

Which leads to:
 * $$Q=V\cdot \rho_{q,0}$$

Which is precisely the result mentioned above for volume charge density. The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.

Discrete charges
If the charge in a region consists of $$N$$ discrete point-like charge carriers like electrons the charge density can be expressed via the Dirac delta function, for example, the volume charge density is: Here, and $$\mathbf r_i$$ the position of the $$i$$th charge carrier. If all charge carriers have the same charge $$q$$ (for electrons $$q=-e$$) the charge density can be expressed through the charge carrier density $$n(\mathbf r)$$: Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.

Quantum charge density
In quantum mechanics, charge density is related to wavefunction $$ \psi(\mathbf r)$$ by the equation


 * $$\rho_q(\mathbf r) = q\cdot|\psi(\mathbf r)|^2 $$

when the wavefunction is normalized as


 * $$Q= q\cdot \int |\psi(\mathbf r)|^2 \, d\mathbf r $$

Application
The charge density appears in the continuity equation which follows from Maxwell's Equations in the electromagnetic theory.