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# Coulomb's law

In physics, Coulomb's law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume (ideally, a point source) exerts on another.

When one is interested only in the magnitude of the force (and not in its direction), it may be easiest to consider a simplified, scalar version of the law

$F = k \frac{\left|q_1 q_2\right|}{r^2}$

where (in SI units):

$F \$ is the magnitude of the force exerted, measured in newtons

$q_1 \$ is the charge on one body, measured in coulombs

$q_2 \$ is the charge on the other body, also measured in coulombs

$r \$ is the distance between them measured in metres

$k \$ is the electrostatic constant or Coulomb force constant, also written as $\frac{1}{ 4 \pi \epsilon_0}$ where $\epsilon_0 \$ is a fundamental physical constant, the permittivity of free space. $k \$ ≈ 8 987 742 438 F−1·m or C−2·N·m2, and $\epsilon_0 \$ ≈ 8.854 &times 10−12 F·m−1 or C2·N−1·m−2. In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb force constant is 1.

Note that $\frac{1}{\mu_0\epsilon_0}=c^2$, where μ0 is the permeability of vacuum and c is the speed of light.

Among other things, this formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each substance and inversely proportional to the square of the distance between them.

The force $F \$ acts on the line connecting the two charged objects.

For calculating the direction and magnitude of the force simultaneously, one will wish to consult the full-blown vector version of the Law

$\mathbf{F} = \frac{1}{ 4 \pi \epsilon_0} \frac{q_1 q_2 \mathbf{r}}{ \left|\mathbf{r}\right|^3}$

where

$\mathbf{F}$ is the electrostatic force vector,

$\mathbf{r}$ is the vector between the two charges, such that

$\mathbf{r}=\mathbf{r_1}-\mathbf{r_2}$

where

$\mathbf{r_1} \$ is vector indicating the position of the charge on which the force acts

$\mathbf{r_2} \$ is the vector indicating the position of the other charge.

This vector equation indicates that opposite charges attract, and like charges repel. When q1q2 is negative, the force is attractive. When positive, the force is repulsive. $|\mathbf{r}|$ has been raised to the third power instead of the second in the denominator in order to normalize the length of the $\mathbf{r}$ vector in the numerator to 1.

Below is a graphical representation of Coulomb's law. $\mathbf{F_2}$ is the force experienced by $\mathbf{Q_2}$. $\mathbf{R_{12}}$ is the vector between two charges ($\mathbf{Q_1}$ and $\mathbf{Q_2}$).

In either formulation, Coulomb's law is fully accurate only when the substances are static (stationary), and remains approximately correct only for slow movement. When movement takes place, magnetic fields are produced that alter the force on the two substances. Especially when rapid movement takes place, the electric field will also undergo a transformation described by Einstein's theory of relativity.

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