📂Electrodynamics

Electric Charge

Definition¹

Charge is one of the fundamental properties of matter and is the source of electromagnetic interactions. It is classified into positive and negative charges; like charges repel each other, and opposite charges attract.

Explanation

Charge is a scalar physical quantity, and since it is divided into positive and negative charges, the charge $q$ of a particle (or object) can take values in the real numbers.

$$ q \in \mathbb{R} $$

Positive charge is exemplified by the proton, and negative charge by the electron. A point charge denotes an idealized charge that has no volume and exists at a single point in space with a net charge. Just as a point in the Euclidean axiomatic system of geometry is an abstract entity without size that specifies only position, a point charge is a theoretical concept without size or volume that specifies only its position and its charge.

The fundamental law of electromagnetism that like charges repel and unlike charges attract is called Coulomb’s law. Consider two particles with charges $q_{1}$ and $q_{2}$, and let the displacement vector between them be $\mathbf{r} = \mathbf{r}_{1} - \mathbf{r}_{2}$. Then the force $\mathbf{F}$ experienced by the particle with charge $q_{1}$ is given by:

$$ \mathbf{F} = \dfrac{1}{4\pi\varepsilon_{0}} \frac{q_1 q_2}{r^{2}} \hat{\mathbf{r}} $$

This force $\mathbf{F}$ is called the Coulomb force or electric force. If the signs of $q_{1}$ and $q_{2}$ are opposite, the direction of the force is toward $q_{2}$ (so $q_{1}$ is attracted); if the signs are the same, $q_{1}$ experiences a force away from $q_{2}$ (i.e., repulsion).

Elementary charge

The unit🔒(26/04/19)unit of charge used in the International System of Units is the coulomb. Because it is named after a person, the symbol is written as an uppercase $\mathrm{C}$, but note that the name of the unit, coulomb, is written with a lowercase initial. Thus Coulomb refers to the French physicist Charles de Coulomb, while coulomb denotes the unit of charge in the International System of Units. The elementary charge denotes the magnitude of charge carried by a single proton or electron, and is denoted by $e$.

$$ e = 1.602\ 176\ 634 \times 10^{-19}\ \mathrm{C} $$

This is not an approximation but an exact value, because the ampere $\mathrm{A} = \mathrm{C} / \mathrm{s}$ itself has been defined as a unit such that $e$ has the above value.

Conservation law

In an isolated system, the total amount of charge is conserved. This is called global charge conservation. The conservation of charge is a fact guaranteed mathematically by Noether’s theorem. Global charge conservation does not, however, guarantee that a charge cannot disappear in one region of the system and simultaneously appear in a distant region. The statement that such nonlocal disappearance/appearance does not occur is local charge conservation, which is expressed mathematically by the continuity equation.

$$ \dfrac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{J} $$

Quantization

Charge is quantized. All charges always exist as integer multiples of the elementary charge.

$$ Q = n e \quad (n \in \mathbb{N}) $$