Electrostatics- Coulomb's Law

1. What is Coulomb's Law?

Coulomb's Law is a fundamental principle in physics that describes the electrostatic force between two stationary, electrically charged particles.

In simple terms:

• Like charges REPEL (e.g., positive and positive, or negative and negative).
• Opposite charges ATTRACT (e.g., positive and negative).

Coulomb's Law tells us exactly how strong this force of attraction or repulsion is.

2. The Formula

The magnitude (strength) of the electrostatic force (F) between two point charges is given by the formula:

F = k |q₁ q₂| / r²

Where:

• F is the electrostatic force (measured in Newtons, N).
• k is Coulomb's constant, which is approximately k ≈ 8.99 × 10⁹ N·m²/C². For most calculations, using k = 9 × 10⁹ is accurate enough.
• q₁ is the magnitude of the first charge (measured in Coulombs, C).
• q₂ is the magnitude of the second charge (measured in Coulombs, C).
• r is the distance between the centers of the two charges (measured in meters, m).

Note on |q₁ q₂|: The absolute value lines mean we only use the positive values of the charges in the formula. This gives us the magnitude (strength) of the force. We determine the direction (attraction or repulsion) by looking at the signs of the charges.

3. Direction of the Force

The formula gives you the size of the force. The direction depends on the signs of the charges:

• If q₁ and q₂ have the SAME sign (both + or both -): The force is REPULSIVE. The charges will push each other apart.
• If q₁ and q₂ have DIFFERENT signs (one + and one -): The force is ATTRACTIVE. The charges will pull each other together.

This force is always mutual. The force that q₁ exerts on q₂ is equal in strength and opposite in direction to the force that q₂ exerts on q₁ (this is an example of Newton's Third Law).

4. Key Relationships (Inverse-Square Law)

Coulomb's Law reveals two critical relationships:

1. Force vs. Charge (Direct Relationship)

• The force is directly proportional to the product of the charges (F ∝ q₁ q₂).
• If you double the size of one charge (e.g., double q₁), the force doubles.
• If you double both charges, the force quadruples (2 × 2 = 4).

2. Force vs. Distance (Inverse-Square Law)

• The force is inversely proportional to the square of the distance (F ∝ 1/r²).
• This is a very important relationship in physics.
• If you double the distance (r → 2r), the force becomes 1/(2r)² = 1/(4r²). The force gets 4 times weaker.
• If you halve the distance (r → r/2), the force becomes 1/(r/2)² = 4/r². The force gets 4 times stronger.

5. Examples

Example 1: Repulsion

A charge q₁ = +2.0 × 10^-6 C is placed 0.03 m (3 cm) away from a second charge q₂ = +3.0 × 10^-6 C. What is the electrostatic force between them?

1. Identify Direction: Both charges are positive, so the force is REPULSIVE.
2. Use the Formula:

F = k |q₁ q₂| / r²
F = (9 × 10⁹ N·m²/C²) |(+2.0 × 10^-6 C)(+3.0 × 10^-6 C)| / (0.03 m)²
F = (9 × 10⁹) (6.0 × 10^-12) / 0.0009
F = (9 × 10⁹) (6.0 × 10^-12) / (9 × 10^-4)
F = (9/9) × 6.0 × 10^{(9 - 12 + 4)}
F = 1 × 6.0 × 10¹ = 60 N

Answer: The force is 60 N, and it is repulsive.

Example 2: Attraction

A charge q₁ = +5.0 × 10^-9 C is placed 0.02 m (2 cm) away from a second charge q₂ = -4.0 × 10^-9 C. What is the electrostatic force between them?

1. Identify Direction: The charges have opposite signs, so the force is ATTRACTIVE.
2. Use the Formula:

F = k |q₁ q₂| / r²
F = (9 × 10⁹ N·m²/C²) |(+5.0 × 10^-9 C)(-4.0 × 10^-9 C)| / (0.02 m)²
F = (9 × 10⁹) (20 × 10^-18) / 0.0004
F = (9 × 10⁹) (20 × 10^-18) / (4 × 10^-4)
F = (9 × 5) × 10^{(9 - 18 + 4)}
F = 45 × 10^-5 N

This can be written in standard scientific notation as F = 4.5 × 10^-4 N.

Answer: The force is 4.5 × 10^-4 N, and it is attractive.

Summary

• Coulomb's Law calculates the electrostatic force between two charges.
• Like charges repel; opposite charges attract.
• The force gets much weaker as the distance increases (an inverse-square law).
• The force gets stronger as the magnitude of the charges increases.