Gauss's theorem & electric flux

In summary, the conversation discusses the application of the divergence theorem involving a point charge enclosed by a Gaussian surface. The author rewrites the expression for ∫sE dot dA as a double integral in spherical coordinates, with limits of integration from 0 to pi and 0 to 2pi. The individual describes difficulty understanding how these rotations integrate over the entire sphere, and provides a visualization using half a circle and a line rotated from 0 to pi, then rotated around the axis of the half circle for a full 2pi revolution. A resource for understanding spherical coordinates is also mentioned.
  • #1
bcjochim07
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Homework Statement



I am reading through my textbook about an application of the divergence theorem involving a point charge enclosed by some arbitrary Gaussian surface. When the author evaluates the ∫sE dot dA, they rewrite the expression as a double integral in spherical coordinates I am fine with this except I can't quite grasp the limits of integration that are given; they are 0 to pi and 0 to 2pi. I am having trouble picturing how these rotations integrate over the whole sphere, as I keep visualizing that both the limits should be 0 2pi. Any suggestions would be greatly appreciated. Thanks.

Homework Equations





The Attempt at a Solution

 
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  • #2
that would cover the surface twice

2 pi is one full revolution

imagine half a circle, swept by a line rotated at its base from 0 to pi

then rotate this about the axis defining by the half circle by full 2 pi to generate a sphere

http://mathworld.wolfram.com/SphericalCoordinates.html
 
  • #3
that would cover the surface twice

2 pi is one full revolution

imagine half a circle, swept by a line rotated at its base from 0 to pi

then rotate this about the axis defining by the half circle by full 2 pi to generate a sphere

http://mathworld.wolfram.com/SphericalCoordinates.html
 
1.

What is Gauss's theorem and how does it relate to electric flux?

Gauss's theorem, also known as Gauss's law, states that the total electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity of free space. Essentially, it relates the electric flux, which is a measure of the strength of an electric field passing through a surface, to the charge enclosed by that surface.

2.

How is electric flux calculated using Gauss's theorem?

Electric flux is calculated by taking the dot product of the electric field and the area vector of the surface through which the field passes. This value can then be multiplied by the surface area to obtain the total electric flux passing through the surface.

3.

What is the significance of Gauss's theorem in electrostatics?

Gauss's theorem is an important tool in understanding the behavior of electric fields and charges in electrostatics. It allows us to calculate electric flux, which is a fundamental quantity in electrostatics, and make predictions about the behavior of electric fields and charges in various situations.

4.

What are the applications of Gauss's theorem in real-life situations?

Gauss's theorem has many practical applications in various fields such as electrical engineering, physics, and chemistry. It is used to calculate the electric field and potential due to different charge distributions, determine the behavior of electric fields in conductors, and analyze the behavior of electric fields in insulators.

5.

What are the limitations of Gauss's theorem?

Gauss's theorem is only applicable to static electric fields and charges. It does not take into account dynamic situations such as changing electric fields or moving charges. Additionally, it is only valid in situations where the charge distribution is symmetric, making it difficult to apply in more complex scenarios.

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