# Coulomb’s Law – the force of interaction between electric charges

From mechanics we know that, between two bodies, which are not in contact with each other, there is a small gravitational attraction force.

Between two objects with electrical charge, depending on the sign of the charges, positive or negative, the interaction force can be of attraction or repulsion.

Let’s consider two electrical charges q1 and q2, separated by the distance r. Between the electrical charges there is an interaction force which is attractive if the charges have opposite signs and repulsive if both charges have the same sign (either positive or negative).

The Coulomb force (F)also called electrostatic force or Coulomb interaction, states that the magnitude of the electrostatic force of interaction between two point electrical charges (q1, q2) is directly proportional to the scalar multiplication of the magnitudes of electrical charge and inversely proportional to the square of the distance (r) between them.

The Coulomb force is along the straight line joining them. If the two electrical charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.

The mathematical expression of Coulomb’s law is:

$\bbox[#FFFF9D]{F = k \frac{q_1 q_2}{r^2}} \tag{1}$

where:

F [N] – Coulomb force
q1, q2 [C] – electrical charges
r [m] – distance between electrical charges
k [F/m] – is called the Coulomb’s constant, or electric force constant or electrostatic constant.

The value of Coulomb’s constant is calculated as:

$k = \frac{1}{4\pi\varepsilon_0} \tag{2}$

where ε0 is the electrical permittivity of free space (vacuum).

The electrical permittivity is a constant, of value:

$\varepsilon_0 = 8.854187817 \cdot 10^{-12} \quad \frac{F}{m}$

The unit of measurement for electrical permittivity is Farad per meter.

Replacing the value of electrical permittivity in equation (2), we can calculate the value of Coulomb’s constant:

$k = 8.987552 \cdot 10^9 \quad \frac{Nm^2}{C^2}$

If the electrical charges are placed in another medium, water for example, instead of using the permittivity of vacuum, we need to use the absolute permittivity ε which is the product between the permittivity of vacuum ε0 and relative permittivity εr.

$\varepsilon =\varepsilon_0 \cdot \varepsilon_r \tag{3}$

In this case, the Coulomb constant will be:

$k = \frac{1}{4\pi\varepsilon} \tag{4}$

Example. Calculate the Coulomb force between two electrons in water, at the distance of 1 mm from each other.

$\begin{equation*} \begin{split} q_1 &= q_2 = 1.6 \cdot 10^{-19} \quad C\\ r &= 0.001 \quad m\\ \varepsilon_0 &= 8.854187817 \cdot 10^{-12} \end{split} \end{equation*}$

The relative permittivity of water at 20 °C is:

$\varepsilon_r = 80.1$

The Coulomb force will be:

$\begin{equation*} \begin{split} F &= \frac{1}{4\pi\varepsilon} \cdot \frac{q_1 q_2}{r^2}\\ &=\frac{1}{4 \cdot \pi \cdot 8.854187817 \cdot 10^{-12} \cdot 80.1} \cdot \frac{1.6 \cdot 10^{-19} \cdot 1.6 \cdot 10^{-19}}{0.001^2}\\ &= 2.872426 \cdot 10^{-24} \quad N \end{split} \end{equation*}$

In order for Coulomb’s law to be valid, several conditions need to be fulfilled:

• the charges must have a spherically symmetric distribution
• the charges must not be in contact
• the charges must be stationary with respect to each other

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