Why does Gauss' law hold for any closed surface?

In summary, Gauss' law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the total enclosed charge. It holds for any closed surface due to the symmetry of electric fields and can be applied to point charges by considering a spherical surface. It can also be used to calculate the electric field of a system of multiple charges and is a consequence of other laws in electromagnetism, such as Coulomb's law and the principle of superposition.
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Why does Gauss' law hold for any closed surface? and can you show this mathematically.

Many thanks :)
 
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Chapter 1 in Classical Electrodynamics by J. D. Jackson provides this derivation.
 
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Physgeek64 said:
can you show this mathematically.
Starting from what assumptions?

Also, a Google search for "proof of Gauss law" turns up a number of pages which may contain what you're looking for, including a number of previous threads here on PF. :oldwink:
 
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1. What is Gauss' law?

Gauss' law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the total enclosed charge.

2. Why does Gauss' law hold for any closed surface?

Gauss' law holds for any closed surface because of the symmetry of electric fields. The electric field lines always originate from positive charges and terminate at negative charges, creating a net flux through a closed surface. This principle is known as the divergence theorem and is the mathematical basis for Gauss' law.

3. How does Gauss' law apply to point charges?

Gauss' law can be applied to point charges by considering a spherical surface surrounding the point charge. The electric flux through this surface is equal to the enclosed charge divided by the permittivity of free space.

4. Can Gauss' law be used to calculate the electric field of a system of multiple charges?

Yes, Gauss' law can be used to calculate the electric field of a system of multiple charges. By choosing a closed surface that encloses all of the charges, the electric flux through this surface can be calculated and used to determine the electric field at any point within the system.

5. Is Gauss' law a consequence of other laws in electromagnetism?

Yes, Gauss' law is a consequence of other laws in electromagnetism, namely the Coulomb's law and the principle of superposition. Coulomb's law describes the force between two charges, and the principle of superposition states that the total electric field at a point is the vector sum of the individual electric fields due to each charge in the system. Together, these laws lead to the mathematical formulation of Gauss' law.

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