Coulomb Law

Coulomb's Law is the foundational principle of electrostatics, describing the force between two stationary point charges. Every analysis in electromagnetic theory, from electric fields to potential distributions, ultimately traces back to this single force relationship. For GATE and university exams, this law and its vector form are tested repeatedly in both direct and applied forms.

Core Concept Explanation

Coulomb's Law states that the force between two stationary point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. The force acts along the line joining the two charges. When the charges have the same sign, the force is repulsive; when they have opposite signs, the force is attractive.

The scalar form of Coulomb's Law is written as F = k |Q1 Q2| / r², where k = 1/(4πε₀) is Coulomb's constant. The value ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space. In a medium with relative permittivity εr, the denominator becomes 4πε₀εr r², which reduces the force compared to vacuum.

The vector form is more complete and is used in field calculations. If charge Q1 is at position r1 and Q2 is at r2, the force on Q2 due to Q1 is given by F21 = (Q1 Q2 / 4πε₀ |r21|²) × r̂21, where r21 = r2 − r1 is the vector from Q1 to Q2 and r̂21 is its unit vector. This form automatically gives direction and sign of force.

Superposition Principle

The superposition principle states that the total force on a charge due to a group of charges is the vector sum of the individual forces exerted by each charge separately. If N charges Q1, Q2, ..., QN act on a test charge Q, the net force is F = sum of all Fi = sum of (Q Qi / 4πε₀ ri²) r̂i. This linear superposition is exact in electrostatics and is heavily used in GATE problems involving charge configurations.

Mathematical Expression

The complete vector Coulomb force on charge Q2 due to charge Q1 placed at the origin is expressed as F = (Q1 Q2 / 4πε₀ r²) r̂, where r is the separation distance and r̂ is the unit vector pointing from Q1 toward Q2. For a medium with relative permittivity εr, the expression becomes F = (Q1 Q2) / (4πε₀ εr r²). The negative of Newton's third law is embedded here: F on Q1 due to Q2 is equal and opposite to F on Q2 due to Q1.

Practical Understanding

Coulomb's Law applies strictly to point charges at rest. In practice, charged metal spheres can be approximated as point charges when the separation is much larger than their size. The inverse-square nature of the force means that doubling the distance reduces the force to one-fourth, which is a common pattern in GATE numerical problems.

Inside a dielectric medium, the permittivity increases by a factor εr (relative permittivity), which reduces the Coulomb force. For example, water with εr ≈ 80 reduces the electrostatic force between charges to about 1/80th of its vacuum value. This has significant implications in chemistry and bioelectricity.

Given:
Q1 = +4 µC = 4×10⁻⁶ C
Q2 = −3 µC = −3×10⁻⁶ C
r = 0.2 m
Medium: free space, ε₀ = 8.854×10⁻¹² F/m

Why this formula applies:
Two point charges in free space — direct application of Coulomb scalar law for magnitude.

Formula:
F = |Q1 × Q2| / (4πε₀ r²)

Substitution:
F = |4×10⁻⁶ × (−3×10⁻⁶)| / (4π × 8.854×10⁻¹² × (0.2)²)
F = 12×10⁻¹² / (4π × 8.854×10⁻¹² × 0.04)

Calculation:
Denominator = 4 × 3.1416 × 8.854×10⁻¹² × 0.04
            = 4.450×10⁻¹²
F = 12×10⁻¹² / 4.450×10⁻¹²

Final Answer:
F ≈ 2.70 N  (attractive, since charges are opposite)

Exam Tip: In GATE, when a problem says 'point charges in free space', always use ε = ε₀ only. If a medium is mentioned, replace ε₀ with ε₀εr. A common trap is forgetting to square r in the denominator when the distance is given in centimetres — always convert to metres first.

• Force between two charges is along the line joining them. Like charges repel; unlike charges attract.
• The magnitude depends on the product of charge magnitudes and decreases as the square of distance increases.
• In a medium with relative permittivity εr, the force is reduced by a factor εr compared to vacuum.
• Superposition allows force calculation in multi-charge systems by treating each pair independently.
• The vector form encodes both magnitude and direction and must be used for field derivation problems.

Quick Revision

• Coulomb's Law: F = |Q1 Q2| / (4πε₀ r²) in free space; replace ε₀ with ε₀εr in a medium.
• Force is a vector — direction given by unit vector r̂ from source charge to field charge.
• Superposition principle: net force = algebraic vector sum of all individual Coulomb forces.
• k = 1/(4πε₀) = 9×10⁹ N·m²/C² in free space.
• Inverse-square law: doubling r reduces F by factor of 4.
• Trap: do not forget to convert distance to metres before substituting.
• Trap: in medium problems, forgetting εr is the most common GATE error in Coulomb force calculations.