Flux integral over a closed surface

In summary, Gauss' law states that the flux integral of an electric field over a closed surface is only dependent on the charge confined within the surface. This is evident for a sphere due to the proportional relationship between the field strength and surface area. However, this law holds true for all closed surfaces, as long as they do not cross any charges. This is because the flux is equal to the divergence of the electric field, which is equivalent to the charge density integrated over the volume. Therefore, as long as the surface does not cross any charges, the total flux will remain the same regardless of its shape or size.
  • #1
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So we recently began electrostatics and here you encounter Gauss' law saying that the flux integral of an electric field E over a closed surface is only dependent on the charge confined within the surface.

Now for a sphere that's pretty obvious why. Because since the field gets weaker proportional to 1/r2 but the area gets bigger proportional to r2 evidently those two things should cancel.

However! It is common knowledge that Gauss' law works for all kinds of surfaces, as long as they are closed. How can I realize that must be true? Because somehow it all hinges on the fact that the area gets bigger proportional to r2, and it is definitely not intuitive for me, that that should be true.
 
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  • #2
I don't think that you can make that kind of sense of gauss law, but you always choose the surface exhibiting the most symmetry in your problem i.e. you make up a surface sourrounding the charge and find the field. And not all fields goes like 1/r^2, how about a line charge exhibiting cylindrical symmetry?
 
  • #3
hmm.. I don't see what you mean. Everything hinges the symmetry of the 1/r2 and you can always break your field up into a superposition of fields that go like 1/r2.

Indeed Gauss' law holds for all closed surfaces.
 
  • #4
Gauss's law is nothing but the divergence theorem which says the flux of a vector field over any closed surface is equal to the divergence of the vector field integrated over the volume. For electric field, the divergence is the charge density, which integrates to total charge enclosed. If you deform your surface from one to another, as long as you don't cross any charges, the total flux over the two surfaces are the same.
 

1. What is a flux integral over a closed surface?

A flux integral over a closed surface is a mathematical concept used in vector calculus to calculate the flow of a vector field through a closed surface. It represents the amount of the vector field that passes through the surface in a given direction.

2. How is a flux integral over a closed surface calculated?

A flux integral over a closed surface is calculated using the formula ∫∫∫S F · dS, where F is the vector field and dS represents a small surface element on the closed surface S. This integral is then evaluated over the entire surface to determine the total flux.

3. What is the significance of a flux integral over a closed surface in physics?

In physics, a flux integral over a closed surface is used to calculate the net flow of a physical quantity, such as electric or magnetic field, through a closed surface. It is an important tool in understanding the behavior of these fields and their interactions with other objects.

4. Can a flux integral over a closed surface be negative?

Yes, a flux integral over a closed surface can be negative. This indicates that the vector field is flowing in the opposite direction of the surface normal, resulting in a net flow out of the surface. This can happen, for example, when the vector field is pointing towards the inside of a closed surface.

5. What are some real-world applications of flux integrals over closed surfaces?

Flux integrals over closed surfaces have numerous applications in fields such as electromagnetism, fluid mechanics, and heat transfer. They are used to calculate the flow of electric and magnetic fields through conductors, the flow of fluids through pipes or channels, and the transfer of heat through different materials.

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