Does Gauss's Law Allow for Non-Enclosed Point Charges on Gaussian Surfaces?

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Discussion Overview

The discussion centers on the application of Gauss's Law in relation to point charges that are not enclosed by a Gaussian surface. Participants explore the implications of choosing different Gaussian surfaces and the resulting electric field behavior, touching on concepts from electrostatics and mathematical interpretations of integrals.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that if a Gaussian surface does not enclose a point charge, the flux through the surface is zero, leading to the conclusion that the electric field must be zero everywhere on that surface, which seems contradictory to Coulomb's Law.
  • Another participant clarifies that while the total flux is zero, the electric field itself is not zero on the surface, using an analogy of water flow to illustrate that there can be non-zero electric field lines even when the net flux is zero.
  • A different analogy is presented, comparing the Gaussian surface to the moon and a charge on Earth, explaining how the electric flux can vary across different points on the surface while still resulting in a net flux of zero.
  • One participant challenges the initial claim that a zero integral implies a zero integrand, emphasizing that a zero definite integral does not necessarily indicate that the integrand is zero everywhere.
  • Another participant introduces the concept of divergence in relation to Gauss's Law, stating that if there is no charge inside a closed surface, the divergence of the electric field is zero, leading to a conclusion about the electric field behavior.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Gauss's Law when applied to Gaussian surfaces that do not enclose charges. There is no consensus on whether the electric field must be zero on such surfaces, as some argue for non-zero fields while others maintain that the flux being zero leads to specific conclusions about the field.

Contextual Notes

Participants highlight limitations in understanding the implications of integrals in the context of Gauss's Law, particularly regarding the relationship between zero flux and the electric field. The discussion also reflects a dependency on the definitions and interpretations of electric field behavior in relation to Gaussian surfaces.

kaotak
Consider a gaussian surface of arbitrary size and a point charge located outside of the gaussian surface at an arbitrary distance.

Gauss's law states that the flux through the gaussian surface is zero, since there is no charge enclosed by that surface. From this we can deduce that the electric field must be zero everywhere on the surface, since the flux is equal to the integral of the dot product of the electric field and dA. But from Coulomb's Law we know that the E = kq/r^2 at any point on the surface, where r is the distance from the point charge.

It seems to matter what gaussian surface you choose and whether or not it encloses the point charge. If you choose a spherical gaussian surface centered around the point charge, you can easily derive E = kq/r^2. So why does this contradiction occur if you choose a gaussian surface that does not enclose the point charge? Are you simply not supposed to or allowed to do that?
 
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The electric field is not zero on the surface! Only the total flux integrated over the surface is zero, indicating there is no source inside. It might help to think of water flowing through the volume. Gauss's law says as much water leaves as enters, since there's no source. That's different from the flow itself, which is nonzero on the surface.
 
Imagine that the Gaussian surface is the surface of the moon, and the (positive) charge is located on the earth. On the side of the moon that faces the earth, the electric flux "goes into" the moon, and is negative; whereas on the far side of the moon, the flux "comes out" of the moon (having "passed through" it), and is positive. The net flux over the entire surface is zero.
 
Last edited:
"From this we can deduce that the electric field must be zero everywhere on the surface, since the flux is equal to the integral of the dot product of the electric field and dA."

A zero integral does not imply a zero integrand.
 
Meir Achuz said:
A zero integral does not imply a zero integrand.
That comment is misleading. A zero indefinite integral implies a zero integrand. However, a zero definite integral does not imply a zero integrand.
 
Gauss's Law can be based on the volume integral of the divergene of the electric field. If there's no charge inside a closed surface, then the divergence of E is zero, and in a couple of simple steps you are done.

Regards, Reilly Atkinson
 

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