PlanetPhysics/Coulomb's Law

Coulomb's law describes the [[../WheatstoneBridge/|electrostatic force]] between [[../Charge/|Electric Charges]]. Coulomb was the first scientist to measure the force using a torsion balance and he discovered that the attraction or repulsion force increases or decreases inversely as the [[../PiecewiseLinear/|square]] of the distance between the charges. As an equation Coulomb's law is given as

$F=k\frac{q_1{q}_2}{r^2}$

This equation only gives the [[../AbsoluteMagnitude/|magnitude]] of the force, but we must also take into account the direction of the force. Similar to gravity, the force between the charges acts along a line between them. Coulomb's law in [[../Vectors/|vector]] form is

$\overrightarrow{F}=k\frac{q_1{q}_2}{r_{12}^3}{\overrightarrow{r}}_{12}$

Some explanation is needed for the variables. $r_{12}$ is the distance between the two charges. The direction of the force is taken into account through the [[../PureState/|unit vector]] ${\hat{r}}_{12}$ which will point towards the other charge or in the oppososite direction depending on attraction or repulsion. So to go from the first equation we add the unit vector

$\overrightarrow{F}=k\frac{q_1{q}_2}{{r^2}_{12}}{\hat{r}}_{12}$

To get a unit vector we divide the vector between the charges by its magnitude

${\overrightarrow{r}}_{12}=r_{12}{\hat{r}}_{12}$

so the unit vector is

${\hat{r}}_{12}=\frac{{\overrightarrow{r}}_{12}}{r_{12}}$

repacing the unit vector with the above equation yields the vector form of Colomb's Law.

More to come on how Coulomb used a torsion balance to come up with this relationship...