Are Gauss's Law and Coulomb's Law Equivalent?

In summary, Gauss's law can be derived from Coulomb's law, but the reverse is not possible. This is because the two forms are not logically equivalent. However, they are essentially the same as they both describe the electric field of a point charge. Gauss's law can be extended to any charge distribution using superposition, and it is commonly introduced in introductory textbooks on electromagnetism as a way to derive Coulomb's law.
  • #1
Enjolras1789
52
2
Gauss's law may be used to derive Coulomb's law. I've never seen it done the other way around. Is this possible? Because I have not seen it, I infer that these forms are not logically equivalent.
 
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  • #2
They are the same. Take a point charge. Since the Coulomb field is everywhere divergenceless except at the position of the charge, you can show it doesn't matter what shape your surface containing the charge is, the flux through any surface containing the charge is the same. You can easily evaluate the flux using a sphere with the charge at the origin. This gives Gauss' law for a point particle, but you can extend it to any charge distribution using superposition.
 
  • #3
Almost every intro textbook in EM (at least the more popular ones) introduce Coulomb Law and use it to derive the Gauss Law. It's usually done for the case for the point-particle, and then generalised to a continuous charge distribution.
 

Related to Are Gauss's Law and Coulomb's Law Equivalent?

1. What is the difference between Gauss's Law and Coulomb's Law?

Both Gauss's Law and Coulomb's Law are fundamental laws of electromagnetism that describe the behavior of electric charges. However, Gauss's Law is a generalization of Coulomb's Law and applies to more complex situations.

2. How does Gauss's Law differ from Coulomb's Law mathematically?

Coulomb's Law states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Gauss's Law, on the other hand, relates the electric flux through a closed surface to the enclosed charge, without explicitly calculating the force between individual charges.

3. Which law is more useful in practical applications?

Coulomb's Law is more commonly used in practical applications, as it allows for the calculation of the force between two point charges. Gauss's Law is more useful in theoretical studies and in solving complex problems, such as those involving non-uniform charge distributions.

4. Can Gauss's Law and Coulomb's Law be used interchangeably?

No, Gauss's Law and Coulomb's Law describe different aspects of the behavior of electric charges and cannot be used interchangeably. They are two separate laws that are both necessary for a complete understanding of electromagnetism.

5. How does Gauss's Law support the principle of superposition in electromagnetism?

Gauss's Law is based on the principle of superposition, which states that the total electric flux through a closed surface is equal to the sum of the electric fluxes due to individual charges. This allows for the calculation of the electric field from multiple charges by simply summing the contributions from each individual charge.

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