Charge density

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In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m⁻³), at any point in a volume.^[1][2][3] Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m⁻²), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m⁻¹), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

Like mass density, charge density can vary with position. In classical electromagnetic theory charge density is idealized as a continuous scalar function of position ${\displaystyle {\boldsymbol {x}}}$, like a fluid, and ${\displaystyle \rho ({\boldsymbol {x}})}$, ${\displaystyle \sigma ({\boldsymbol {x}})}$, and ${\displaystyle \lambda ({\boldsymbol {x}})}$ are usually regarded as continuous charge distributions, even though all real charge distributions are made up of discrete charged particles. Due to the conservation of electric charge, the charge density in any volume can only change if an electric current of charge flows into or out of the volume. This is expressed by a continuity equation which links the rate of change of charge density ${\displaystyle \rho ({\boldsymbol {x}})}$ and the current density ${\displaystyle {\boldsymbol {J}}({\boldsymbol {x}})}$.

Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. A charge distribution is ultimately composed of individual charged particles separated by regions containing no charge.^[4] For example, the charge in an electrically charged metal object is made up of conduction electrons moving randomly in the metal's crystal lattice. Static electricity is caused by surface charges consisting of electrons and ions near the surface of objects, and the space charge in a vacuum tube is composed of a cloud of free electrons moving randomly in space. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. However, because the elementary charge on an electron is so small (1.6⋅10⁻¹⁹ C) and there are so many of them in a macroscopic volume (there are about 10²² conduction electrons in a cubic centimeter of copper) the continuous approximation is very accurate when applied to macroscopic volumes, and even microscopic volumes above the nanometer level.

At even smaller scales, of atoms and molecules, due to the uncertainty principle of quantum mechanics, a charged particle does not have a precise position but is represented by a probability distribution, so the charge of an individual particle is not concentrated at a point but is 'smeared out' in space and acts like a true continuous charge distribution.^[4] This is the meaning of 'charge distribution' and 'charge density' used in chemistry and chemical bonding. An electron is represented by a wavefunction ${\displaystyle \psi ({\boldsymbol {x}})}$ whose square is proportional to the probability of finding the electron at any point ${\displaystyle {\boldsymbol {x}}}$ in space, so ${\displaystyle |\psi ({\boldsymbol {x}})|^{2}}$ is proportional to the charge density of the electron at any point. In atoms and molecules the charge of the electrons is distributed in clouds called orbitals which surround the atom or molecule, and are responsible for chemical bonds.