Coulomb's Law: The Electrostatic Force

Introduction to Coulomb's Law

Coulomb's Law states that the electrostatic force (\(F\)) between two 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 formula is given by:

\[ F = k_e \frac{q_1 q_2}{r^2} \]

Where:

F is the magnitude of the electrostatic force between the charges.
q₁ and q₂ are the magnitudes of the two charges.
r is the distance between the two charges.
kₑ is Coulomb's constant, which has a value of \(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\).

Properties of the Electrostatic Force

Several key properties emerge from Coulomb's Law:

If the charges are of the same sign (both positive or both negative), the force is repulsive, meaning the charges push away from each other.
If the charges are of opposite signs, the force is attractive, meaning the charges pull towards each other.
The force increases as the magnitude of the charges increases and decreases as the distance between the charges increases.

Direction of the Force

The direction of the force is determined by the nature of the charges:

If both charges are positive, the force is repulsive, and they push away from each other.
If one charge is positive and the other is negative, the force is attractive, and the charges will pull towards each other.

Example 1: Repulsive Force Between Two Positive Charges

Consider two point charges, both with a charge of \(q_1 = 2 \, \mu C = 2 \times 10^{-6} \, C\) and \(q_2 = 3 \, \mu C = 3 \times 10^{-6} \, C\). The distance between them is \(r = 0.5 \, \text{m}\).

The force between these two charges can be calculated using Coulomb's Law:

\[ F = k_e \frac{q_1 q_2}{r^2} \]

Substituting the values:

\[ F = \left( 8.99 \times 10^9 \, \frac{\text{N} \cdot \text{m}^2}{\text{C}^2} \right) \frac{(2 \times 10^{-6} \, \text{C}) (3 \times 10^{-6} \, \text{C})}{(0.5 \, \text{m})^2} \]

Solving the equation:

\[ F = (8.99 \times 10^9) \times \frac{6 \times 10^{-12}}{0.25} = 2.16 \, \text{N} \]

Thus, the electrostatic force between the two charges is \(2.16 \, \text{N}\), and since both charges are positive, the force is repulsive.

Example 2: Attractive Force Between Opposite Charges

Now consider two point charges: \(q_1 = 5 \, \mu C = 5 \times 10^{-6} \, C\) and \(q_2 = -3 \, \mu C = -3 \times 10^{-6} \, C\), placed 0.2 meters apart.

We can again apply Coulomb's Law to calculate the magnitude of the force:

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

Substituting the values:

\[ F = \left( 8.99 \times 10^9 \, \frac{\text{N} \cdot \text{m}^2}{\text{C}^2} \right) \frac{|(5 \times 10^{-6} \, \text{C}) (-3 \times 10^{-6} \, \text{C})|}{(0.2 \, \text{m})^2} \]

Solving the equation:

\[ F = (8.99 \times 10^9) \times \frac{15 \times 10^{-12}}{0.04} = 3.37 \, \text{N} \]

Thus, the magnitude of the attractive force between the charges is \(3.37 \, \text{N}\), and since the charges are opposite, the force is attractive.

Superposition Principle

When multiple charges are present, the total electrostatic force on any given charge is the vector sum of all the individual forces exerted by other charges. This is known as the superposition principle. Mathematically, the total force is given by:

\[ \mathbf{F}_\text{total} = \sum_{i} \mathbf{F}_i \]

Conclusion

Coulomb's Law is a fundamental law of electrostatics that governs the interaction between charged particles. It helps explain a wide variety of phenomena, from the behavior of charged objects to the structure of atoms and molecules. Understanding Coulomb's Law is essential in fields like electrical engineering, chemistry, and physics.