Does Gauss' Law use line integrals or surface integrals?

In summary, Gauss' Law can be written in two different forms, but both are equivalent as they are both surface integrals. The only difference is that one form uses a dot product for a more general approach, while the other assumes the electric field is normal to the surface. Some textbooks use a single integral to simplify the notation.
  • #1
Albert Tran
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In my physics textbook, I see Gauss' Law as [PLAIN]https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png[itex]\oint[/itex][itex]E [Broken] dA[/itex] but in other places I see it as [PLAIN]https://upload.wikimedia.org/math/0/3/5/035b153014908c0431f00b5ddb60c999.png[PLAIN]https://upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png[PLAIN]https://upload.wikimedia.org/math/7/f/f/7ff140fff7dde71951767d28cb5304ac.png [Broken] [PLAIN]https://upload.wikimedia.org/math/3/9/f/39f4ca19d0c263fd02c0e50cb8829239.png, [Broken] which one is the right one?
 
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  • #2
The two expressions you wrote are the same, both are surface integrals as you can see from the area element ##dA##, it's just different notation if you want to write out double integral signs or not.

The only difference is in the second one the dot product is used making it more general since we want the component of the ##E## field normal to the surface to get the flux. The first form assumes that ##E## is normal to the surface so we only need to care about the scalar value.
 
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  • #3
Both are the same.. Some books write a single integral to make it less cumbersome.. If you see a "S" underneath the integral sign or the infinitesimal element is dA or dS it's understood to be a surface integral.
 

1. What is Gauss' Law?

Gauss' Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It is one of Maxwell's equations and is used to describe the behavior of electric fields.

2. Does Gauss' Law use line integrals or surface integrals?

Gauss' Law uses surface integrals. This is because it relates the electric flux through a closed surface to the charge enclosed by that surface, which can be calculated using a surface integral.

3. How is Gauss' Law related to Coulomb's Law?

Gauss' Law is an extension of Coulomb's Law. While Coulomb's Law relates the force between two point charges, Gauss' Law relates the total electric field from a charge distribution to the enclosed charge.

4. Can Gauss' Law be applied to all types of surfaces?

Yes, Gauss' Law can be applied to any closed surface, regardless of its shape or orientation. This is because the surface integral can be calculated using the divergence theorem, which states that the surface integral of a vector field is equal to the volume integral of its divergence.

5. How is Gauss' Law used in practical applications?

Gauss' Law is used in many practical applications, such as in the design of electrical circuits and in the analysis of electric fields in various systems. It is also used in the study of electric shielding and in the calculation of capacitance.

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