1: Coulomb's Law

Section 1: Electrostatics

1: Coulomb's Law

Coulomb's Law is the fundamental principle governing the electrostatic force between two stationary point charges. Formulated by Charles-Augustin de Coulomb in 1785, it states:
The magnitude of the electrostatic force ($F$) between two point charges ($q_1$ and $q_2$) is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance ($r$) between them. The force acts along the line joining the charges.

Mathematical Formulation

The scalar form is:
$F = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}$
where:

• $\epsilon_0$ is the permittivity of free space ($8.854 \times 10^{-12} \text{C}^2 \text{N}^{-1} \text{m}^{-2}$),
• $r$ is the separation in meters (m),
• $q_1$ and $q_2$ are the charges in coulombs (C).

The vector form is crucial for direction and superposition:
$\vec{\mathbf{F}}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{\mathbf{r}}_{12}$
Here, $\vec{\mathbf{F}}_{12}$ is the force on charge $q_1$ due to $q_2$, and $\hat{\mathbf{r}}_{12}$ is the unit vector pointing from $q_2$ to $q_1$. The force is:

• Repulsive for like charges ($q_1 q_2 > 0$),
• Attractive for unlike charges ($q_1 q_2 < 0$).

The Superposition Principle

For a system of $n$ point charges, the net force on any charge $q_i$ is the vector sum of forces exerted by all other charges:
$\vec{\mathbf{F}}_{\text{net}} = \sum_{j \neq i} \frac{1}{4\pi\epsilon_0} \frac{q_i q_j}{r_{ij}^2} \hat{\mathbf{r}}_{ij}$
This principle allows calculation of forces in complex charge configurations.

Key Comparisons & Notes

1. Inverse-Square Dependence: Similar to Newton’s Law of Gravitation, but electrostatic force can be attractive or repulsive.
2. Strength: Electrostatic force is $\sim 10^{39}$ times stronger than gravity for protons.
3. Limitations: Applies strictly to point charges (size $\ll r$) and static charges (no motion).
4. Constant: The term $\frac{1}{4\pi\epsilon_0}$ (Coulomb's constant, $k$) equals $9 \times 10^9 \text{N·m}^2 \text{C}^{-2}$.

Understanding Coulomb’s Law is essential for analyzing electric fields, Gauss’s Law, and potential in subsequent sections.