Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two point charges. It was named after the French physicist Charles-Augustin de Coulomb, who first published the law in 1785. Coulomb's Law states that the magnitude of the electrostatic force of interaction between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.

The Law

The mathematical expression for Coulomb's Law is:

$$ F = k_e \frac{|q_1 q_2|}{r^2} $$

where:

• $F$ is the magnitude of the electrostatic force between the two charges.
• $q_1$ and $q_2$ are the magnitudes of the charges.
• $r$ is the distance between the centers of the two charges.
• $k_e$ is Coulomb's constant, which has a value of approximately $8.9875 \times 10^9 \, \text{N m}^2/\text{C}^2$ in vacuum.

The direction of the force is along the line joining the charges, and its sense depends on the nature of the charges. Like charges repel each other, while unlike charges attract each other.

Important Points and Differences

Aspect	Description
Nature of Force	The force can be attractive or repulsive depending on whether the charges are of opposite sign or the same sign, respectively.
Vector Quantity	Coulomb's force is a vector quantity, having both magnitude and direction.
Medium Dependency	The value of $k_e$ can change depending on the medium between the charges. In other mediums, it is often written as $\frac{1}{4\pi\epsilon}$, where $\epsilon$ is the permittivity of the medium.
Applicability	Coulomb's Law is accurate for point charges and also applies to charged objects as long as the charge distribution is spherically symmetric.
Inverse Square Law	The force varies inversely with the square of the distance between the charges, similar to the gravitational force law.
Superposition Principle	When a charge experiences forces from multiple other charges, the total force is the vector sum of the individual forces.

Examples

Example 1: Force Between Two Point Charges

Calculate the force between two charges of $+3 \mu C$ and $-2 \mu C$ placed $0.5 m$ apart in vacuum.

Solution:

Given:

• $q_1 = +3 \mu C = 3 \times 10^{-6} C$
• $q_2 = -2 \mu C = -2 \times 10^{-6} C$
• $r = 0.5 m$
• $k_e = 8.9875 \times 10^9 \, \text{N m}^2/\text{C}^2$

Using Coulomb's Law:

$$ F = k_e \frac{|q_1 q_2|}{r^2} = 8.9875 \times 10^9 \frac{|3 \times 10^{-6} \cdot -2 \times 10^{-6}|}{(0.5)^2} $$

$$ F = 8.9875 \times 10^9 \frac{6 \times 10^{-12}}{0.25} $$

$$ F = 8.9875 \times 10^9 \times 24 \times 10^{-12} $$

$$ F = 215.7 \times 10^{-3} N $$

$$ F = 0.2157 N $$

The force is attractive since the charges are of opposite signs.

Example 2: Superposition Principle

Three point charges are located at the corners of an equilateral triangle with sides of length $0.3 m$. The charges are $+2 \mu C$, $+2 \mu C$, and $-2 \mu C$. Calculate the net force on the $-2 \mu C$ charge.

Solution:

The force on the $-2 \mu C$ charge due to each of the $+2 \mu C$ charges can be calculated using Coulomb's Law. Since the triangle is equilateral, the forces will be equal in magnitude but will have different directions.

Let's calculate the magnitude of the force from one of the $+2 \mu C$ charges:

$$ F = k_e \frac{|q_1 q_2|}{r^2} = 8.9875 \times 10^9 \frac{|2 \times 10^{-6} \cdot -2 \times 10^{-6}|}{(0.3)^2} $$

$$ F = 8.9875 \times 10^9 \frac{4 \times 10^{-12}}{0.09} $$

$$ F = 8.9875 \times 10^9 \times 44.44 \times 10^{-12} $$

$$ F = 399.44 \times 10^{-3} N $$

$$ F = 0.39944 N $$

Since there are two such forces acting at an angle of $120^\circ$ to each other (because of the equilateral triangle), we need to add them vectorially to find the net force. The net force will be along the bisector of the angle between the two forces due to symmetry.

The magnitude of the net force $F_{net}$ can be found using trigonometry:

$$ F_{net} = 2 F \cos(60^\circ) = 2 \times 0.39944 \times \frac{1}{2} $$

$$ F_{net} = 0.39944 N $$

The net force on the $-2 \mu C$ charge is $0.39944 N$ directed away from the center of the triangle.