Coulomb's Law

Coulomb's law is one of the most fundamental and oldest rules of electrostatics. It was experimentally found by Charles Augustine de Coulomb in 1785 and confirmed many times in the following decades. The original experimental setup consists of a torsion balance that can be described as follows: A metal ball is connected to a bar that is attached to a thin wire. If another charged metal ball is brought close to it, the bar rotates until the restoring torque of the twisted wire compensates for the applied torque. When a beam of light is reflected at a mirror connected to the wire, the rotation of the wire can be measured with the help of a scale at a far distance. From the measured value, the force between the two balls can be derived.

Coulomb found out that the force between both balls is proportional to the charges $q$ and $Q$ in each ball and inversely proportional to the squared distance $r$ between them: $$F = k\frac{qQ}{r^2}$$ The constant $k$ is used as a proportional constant. This equation looks identical to Newton's law of gravity if the charges $q$ and $Q$ are placed by masses $m$ and $M$ of two objects. Another important result is that the force is independent of the ball size. We can therefore state that Coulomb's law is also valid for point charges such as electrons or small objects like protons.

The constant $k$ cannot be found in most of the textbooks about electrodynamics, because for historical reasons, it is replaced by $1/(4\pi \varepsilon_0)$ with the electric field constant $\varepsilon_0$. We can formulate Coulomb's law then as follows: $$F_C = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r^2}$$ The electric field constant has the value $\varepsilon_0=8.854\cdot 10^{-12}({\mathrm{A}\,\mathrm{s}})/({\mathrm{V}\,\mathrm{m}})$. The units Ampere (A) and Volt (V) will be introduced in one of the later sections. We already know that the electric field lines have to point away from the center in case of a positively charged object or towards the center for a negatively-charged one. We can therefore rewrite this very important formula using vector notation, similar to Newton's law of gravity: $$\boxed{\vec{F}_C = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r^2}\hat{\vec{r}}}$$ Here, $\hat{\vec{r}} = \vec{r}/r$ denotes the unit vector in radial direction. This is the complete formula as mentioned in most of the textbooks.

Exercises:

• exercises:newton_coulomb (+)