\(\)

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

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 $$ \vec F_2 = \vec F_{1,2} + \vec F_{3,2} \nonumber $$

## Electric vs. gravitational force

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. $$ \left. \begin{align*} F_{el} &= \frac{(400\times 10^3)^2\ 9 \times 10^9}{d^2} \\ F_{gr} &= \frac{6\times 10^{24}\ 6.4\times 10^{23}\ 6.7\times 10^{-11}}{d^2} \end{align*} \right\} \Rightarrow \frac{F_{el}}{F_{gr}} \approx 10^{-17} \nonumber $$