Coulomb's law and Gauss's law are both fundamental principles in electrostatics, and while they are related, they approach the concept of electric fields from different angles. Gauss's law, however, is not directly derived from Coulomb's law alone. Instead, it relies on an additional assumption.
Understanding the Two Laws
To understand their relationship, let's briefly examine each law:
Coulomb's Law
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Coulomb's Law describes the force between two point charges.
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It states that the electric force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
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Mathematically, the force (F) is given by:
F = k * |q1 * q2| / r^2Where:
- k is Coulomb's constant
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
Gauss's Law
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Gauss's Law relates the electric field to the electric charge distribution.
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It states that the total electric flux through any closed surface is proportional to the enclosed electric charge.
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Mathematically, it's expressed as:
∮ E ⋅ dA = Q_enclosed / ε₀
Where:
- ∮ E ⋅ dA is the electric flux through the closed surface.
- Q_enclosed is the net charge enclosed by the surface.
- ε₀ is the permittivity of free space.
The Relationship Explained
The key to understanding the connection between Coulomb's Law and Gauss's Law lies in the superposition principle.
Superposition Principle
- The superposition principle states that the net electric field at a point due to a system of charges is the vector sum of the individual electric fields created by each charge at that point.
- Coulomb's law gives the electric field due to a single point charge only.
- According to the Wikipedia reference, Gauss's Law can be proven using Coulomb’s law if we also assume the electric field obeys the superposition principle.
Building the Bridge:
- Coulomb's Law to Electric Field: Coulomb’s law enables us to calculate the electric field generated by a single point charge.
- Superposition to Multiple Charges: The superposition principle lets us determine the electric field from multiple point charges by vector addition of each charge's individual field calculated using Coulomb's law.
- Gauss's Law Application: Gauss's Law, using the combined electric field from the superposition of the individual electric fields derived via Coulombs law, then allows us to calculate the flux through a closed surface based on the total enclosed charge. It doesn't rely directly on Coulomb’s law but on its application alongside superposition.
Key Differences:
| Feature | Coulomb's Law | Gauss's Law |
|---|---|---|
| Purpose | Calculates force between two point charges | Relates electric flux through a closed surface to the enclosed charge |
| Focus | Interactions of charges | Electric field behaviour related to enclosed charge |
| Calculations | Calculates force/field directly, for single point charge | Calculates flux and indirectly helps find electric field via symmetry considerations |
| Applicability | Directly applicable for point charges | Powerful for systems with high symmetry but can be cumbersome for complex ones. |
| Derivation | Does not need an extra assumption | Requires superposition principle as an additional assumption besides Coulomb's law. |
In essence:
- Coulomb’s law provides the foundation for understanding the force and electric field due to a point charge.
- Gauss’s Law, along with the superposition principle, provides a powerful tool for calculating the electric flux and, often indirectly, the electric field, particularly in situations with high symmetry.
