Derivation of gauss's law for electrical fields

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SUMMARY

The discussion centers on the derivation of Gauss's Law for electric fields, specifically addressing the confusion surrounding the application of the formula \( \Phi = \frac{q}{\varepsilon_0} \) to all closed surfaces. It is established that the electric flux through a closed surface is independent of the surface's shape or size, as long as it encloses the same charge. Participants suggest calculating electric flux for spheres of varying radii using the integral \( \Phi = \int \vec{E} \cdot d\vec{A} \) to gain a deeper understanding of the concept.

PREREQUISITES
  • Understanding of electric flux and its mathematical representation
  • Familiarity with Gauss's Law and its implications in electromagnetism
  • Basic knowledge of electric fields and charge distributions
  • Proficiency in calculus, particularly in evaluating surface integrals
NEXT STEPS
  • Calculate electric flux for spheres of different radii using \( \Phi = \int \vec{E} \cdot d\vec{A} \)
  • Explore the concept of electric field lines and their relation to electric flux
  • Investigate the application of Gauss's Law to non-spherical closed surfaces
  • Study the limits and approximations of closed surfaces using spherical and annular sections
USEFUL FOR

Students and professionals in physics, particularly those studying electromagnetism, as well as educators seeking to clarify the principles of Gauss's Law and electric flux.

Ragnar
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This is a very stupid question. extremely stupid. In fact I'm extremely embarassed.

I was reading a text on electromagnetism, and it said that since the flux due to a charge does not depend on the radius of the sphere then the formula, q/permitivitty applies to all closed surfaces. This is where i got confused. why does that make it apply to all closed surfaces?

please don't laugh at me.

Please I need an answer!
 
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Try calculating the electric flux of spheres with various radii using [itex]\phi = \int\vec{E}\cdot d\vec{A}[/itex] and see where that gets you. Intuitively, one can consider the definition of electric flux (number of electric field lines passing through a unit area) and the definition of the electric field and how it relates to the density of electric field lines...
 
After calculating for spheres, try (say) a "northern" hemisphere at radius r1 and a "southern" hemisphere at radius r2 with an http://mathworld.wolfram.com/Annulus.html" joining their "equators". Note the flux through this annulus.

Then, approximate ANY closed surface by portions of spheres and portions of annuli. ...take limits.
 
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