Coulomb’s Law - Scalar Form of Coulomb’s Law

Statement

Coulomb’s law states that the electrostatic force between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The force acts along the line joining the two charges and is repulsive for like charges and attractive for unlike charges.

Explanation/Mathematical Form

Let two point charges q₁​ and q₂​ be placed at a distance r apart in vacuum (or air).

According to Coulomb’s law:

F ∝ q₁q₂

F ∝ \[\frac {1}{r^2}\] ​

Combining the above relations:

F = k\[\frac {q_1q_1}{r^2}\]​​

where
F = electrostatic force between the charges,
r = distance between the charges,
k = proportionality constant.

In vacuum (or air),

k = 9.0 × 10⁹ N m²C⁻²

Hence,

F = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}\]​​

where ε₀​ is the permittivity of free space, given by

ε₀ = 8.85 × 10⁻¹² C²N⁻¹m⁻²

If the charges are placed in a dielectric medium of permittivity ε,

F = \[\frac{1}{4\pi\varepsilon}\frac{q_1q_2}{r^2}\]​​

and since ε = Kε_0​,

F = \[\frac{1}{4\pi K\varepsilon_0}\frac{q_1q_2}{r^2}\]​​

where K is the dielectric constant of the medium.

Conclusion

Coulomb’s law quantitatively describes the force of attraction or repulsion between two point charges.
The force:

• depends on the magnitudes of charges,
• varies inversely as the square of the distance,
• acts along the line joining the charges, and
• decreases in a dielectric medium by a factor equal to its dielectric constant.