Coulomb’s Law – Statement, Formula, Limitations & Applications Explained

In this guide, you’ll discover everything about Coulomb’s Law — one of the cornerstones of electrostatics. We’ll cover its statement, formula, derivation, limitations, and applications in a clear, simple way so you can understand how electric charges interact in real life.

What is Coulomb’s Law?

Coulomb’s Law describes how electric charges exert force on each other.
Proposed by Charles Augustin de Coulomb in 1785, the law states that the electrostatic force between two stationary point charges is:

• Directly proportional to the product of their charges.
• Inversely proportional to the square of the distance between them.

Because the force decreases rapidly as the distance increases, it is also called the inverse square law in electrostatics.

Coulomb’s experiments with a torsion balance laid the foundation for understanding electric and magnetic fields — concepts that later led to Maxwell’s equations.

Statement of Coulomb’s Law

”The electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance separating them.”

Mathematically,

When we include the proportionality constant k, the equation becomes:

Coulomb’s Law Formula and Derivation

Imagine two point charges Q₁ and Q₂ separated by a distance d.

According to Coulomb’s Law, the force between them is,

• The electrostatic force is directly proportional to the product of charges

• The electrostatic force is inversely proportional to the square of the distance between the centers of the charged bodies, i.e.

Combining these two equations, we get,

Where k is a constant of proportionality whose value depends on the medium in which the bodies are placed.

In the SI system, the value of k is given by,

Where,

• ε₀=8.85×10⁻¹² C²/N⋅m²=8.85×10−12C2/N⋅m2 → permittivity of free space
• ε_r → relative permittivity of the medium

Therefore, the formula of the electrostatic force becomes,

When bodies are separated from each other in a vacuum, then εr=1, thus,

Key Notes:

• If both charges are like charges (positive–positive or negative–negative), the force is repulsive.
• If the charges are unlike (positive–negative), the force is attractive.
• The force always acts along the line joining the two charges.

Coulomb’s Law Formula in Vector Form

If two point charges Q₁ and Q₂ are separated by a distance d, the force on Q₂ due to Q₁​ is:

Where:

• F_21​ → Force on charge Q_2​ due to Q₁
• ε0→ Permittivity of free space (8.854 × 10⁻¹² C²/N·m²)
• d → Distance between the charges
• d^{^}2​1 → Unit vector pointing from Q₁ to Q₂

Key Points:

• Force magnitude: F=1/4πε₀(Q₁Q₂/d²)
• Direction: Along the line joining the charges
• Repulsive if charges are alike, attractive if charges are opposite

Generalized Vector Form:

Limitations of Coulomb’s Law

Coulomb’s Law is accurate under specific conditions. Below are its major limitations-

1. Applicable only for point charges: The law assumes charges are point-sized; it fails for large or irregularly shaped charged bodies.
2. Valid only for stationary charges: It cannot be applied to moving charges, since magnetic effects aren’t considered.
3. Fails in non-uniform media: Coulomb’s Law assumes a uniform medium; it’s inaccurate in materials where permittivity changes.
4. Not accurate for atomic or planetary distances: At very small or very large scales, other forces dominate.

In simple terms, Coulomb’s Law is reliable only for small, stationary point charges placed in a vacuum or uniform dielectric medium.

Applications of Coulomb’s Law

Coulomb’s Law is widely used in physics, engineering, and electrical systems. Here are its key applications:

1. To calculate electrostatic force:
   It determines the magnitude and direction of force between two point charges.
2. To measure the distance between charges:
   Using known charge values and measured force, distance between charges can be calculated.
3. To determine net electrostatic force on a charge:
   When multiple charges act on a single charge, the resultant force can be computed using vector addition based on Coulomb’s Law.
4. In electric field and potential calculations:
   The law serves as the foundation for deriving the electric field intensity and potential due to a charge distribution.
5. In designing capacitors and insulating materials:
   Engineers use Coulomb’s Law to understand how dielectric materials affect electric fields and energy storage.

Example Problems

Example 1 – Calculate the electrostatic force between two charges 7 × 10^-8 C and 2 × 10^-7 C. These two charges are separated by a distance of 15 cm in free space.

Solution – Given data,

According to Coulomb’s law, we have,

Example 2 – Two spheres are separated by a distance of 4 cm in air. The electrostatic charge on one sphere is 5 × 10^-9 C and the electric force of attraction of 5 mN is acting between the spheres. Determine the electric charge on the second sphere.

Solution – Given data,

According to Coulomb’s law, we have,

Conclusion

Coulomb’s Law provides the basic understanding of how electric charges interact.
Although it has some limitations, it remains a fundamental principle for studying electrostatics, designing capacitors, and calculating electric fields and potentials.

By mastering this law, you build the foundation to understand Gauss’s Law, Electric Field, and Electrostatic Potential — the next steps in learning electricity and magnetism.

Frequently Asked Questions (FAQs) on Coulomb’s Law

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