# Coulomb’s law



My notes of the excellent lecture 1 by “Walter Lewin. 8.02 Electricity and Magnetism. Spring 2002. The material is neither affiliated with nor endorsed by MIT, https://youtube.com. License: Creative Commons BY-NC-SA.”

The electric force between two particles decreases with the inverse square of the distance, just as the gravitational force.

If we take two charges $$q_1$$ and $$q_2$$, separated by a distance $$r$$. Then the force $$\vec F_{1,2}$$ from $$q_1$$ on $$q_2$$ is visualized as Force $$\vec F_{1,2}$$ from $$q_1$$ on $$q_2$$, and visa versa

Charles-Augustin de Coulomb, a French physicist, in 1785 published the following relationship $$\shaded{ \vec F_{1,2} = \frac{q_1\,q_2\,K}{r^2}\,\hat r_{1,2} } ,\quad K=\frac{1}{4\pi\varepsilon_0} \tag{Coulomb’s law}$$

Where

• $$\vec F_{1,2}$$, force from $$q_1$$ on $$q_2$$ $$[\rm N]$$
• $$q_1$$ and $$q_2$$, the values of charge $$[\rm C]$$, the unit is named after Coulomb
• $$r$$, distance between the charges $$[\rm m]$$
• $$K$$, Coulomb’s constant $$\approx 9\times 10^9$$
• $$\varepsilon_0$$ is the permittivity of free space.

Note

• The relationship is sign sensitive. If once charge is negative and the other positive the force is in the opposite direction.
• There is a clear parallel with gravity (except that gravity never repels), where $$\vec F_g = \frac{m_1\,m_2\,G}{r^2} \nonumber$$

## Superposition principle

With multiple charges, the superposition principle applies, because it is consistent with all our observations.

Let $$\vec F_{1,2}$$ be the force from $$q_1$$ on $$q_2$$, and $$\vec F_{3,2}$$ be the force from $$q_3$$ on $$q_2$$. Assume $$q_1$$ and $$q_2$$ are positive, and $$q_3$$ is negative. The forces can be shown as

The net force on $$q_2$$ is the vectoral sum of the individual component forces $$\vec F_2 = \vec F_{1,2} + \vec F_{3,2} \nonumber$$
E.g. charge of a proton/electron $$q_{p^+} = q_{e^-} = 1.6\times 10^{-19}\,\rm C \nonumber$$
In our immediate surroundings, electrical forces are much more powerful than gravitational forces. E.g. two protons repel each other with the forces \left. \begin{align*} F_{el} &= \frac{(1.6\times 10^{-19})^2\ 9\times 10^9}{d^2} \\ F_{gr} &= \frac{(1.7\times 10^{-27})^2\ 6.7\times 10^{-11}}{d^2} \end{align*} \right\} \Rightarrow \frac{F_{el}}{F_{gr}} \approx 10^{36} \nonumber (The nuclear forces hold the protons together)
On the scale of planets, it is gravity that holds our world together. Because the large objects have a very small charge per unit mass. \left. \begin{align*} F_{el} &= \frac{(400\times 10^3)^2\ 9 \times 10^9}{d^2} \\ F_{gr} &= \frac{6\times 10^{24}\ 6.4\times 10^{23}\ 6.7\times 10^{-11}}{d^2} \end{align*} \right\} \Rightarrow \frac{F_{el}}{F_{gr}} \approx 10^{-17} \nonumber