The Two forms of Maxwell's equations

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Discussion Overview

The discussion revolves around the relationship between the integral and differential forms of Maxwell's equations, specifically focusing on the concepts of divergence and curl. Participants explore their understanding of these mathematical operators and how they relate to physical interpretations in electromagnetism.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant expresses confusion about the relationship between the integral form of electric field flux and the differential form involving divergence, questioning how they can be equivalent.
  • Another participant clarifies that the integral of E dot dA represents the flux through a surface, which is related to the divergence but not equal to it. They explain that divergence represents the flux through an infinitesimal closed surface.
  • A third participant acknowledges their misunderstanding regarding the definition of divergence as being associated with an infinitesimal closed surface rather than an open point, and they relate this to the concept of curl.
  • A later reply encourages participants to seek multiple interpretations and revisit explanations to deepen their understanding of these concepts.

Areas of Agreement / Disagreement

Participants show varying levels of understanding and confusion regarding the concepts of divergence and curl. While some clarify their misconceptions, there is no consensus on the best way to visualize or interpret these mathematical constructs.

Contextual Notes

Participants mention the importance of understanding the definitions of divergence and curl in the context of Maxwell's equations, highlighting that these concepts may not be immediately clear and require careful study.

Who May Find This Useful

This discussion may be useful for students or individuals seeking to understand the mathematical foundations of electromagnetism, particularly those grappling with the concepts of divergence and curl in relation to Maxwell's equations.

DeepSeeded
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Hello,

I was tempted to put this in the math section but it is more of a visualization problem though it is most likley due to my lack of understanding the divergence and curl operators fully.

I am comfortable with the closed loop integral of E dot dA and can visualize it as a solid closed surface. However when I think of the divergence of the E vector I think of a tiny little piece of the E vector.

How can these be the same thing?

Same goes for the Curl vector being the same as the closed loop integral of a line.

In 8.02 they used the integral form, in 8.03 they are using the differential form and now I am confused.
 
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The integral of the E dot dA is not a surface but the flux through that surface. You can visualize it as the "amount of field" crossing the surface. This is not the same as the divergence but same as the integral of the divergence over the volume included in the surface.
The divergence is the flux through a tiny, infinitesimal, closed surface (divided by the volume inside the surface). By integration of the divergence over the volume you just add all these fluxes inside the volume included by the surface and end up with the "total" (net) flux through the surface.

You can write Maxwell equations in differential or integral form. It does not follow that the flux is equal to the divergence.
 
OK, my misunderstanding was that the definition of the divergence is an infinitsmal sphere which is closed and not an open point. I see that it says this now on wikipedia but it was not so clear.

Looks like the definition of the curl is an infinitsmal closed line integral which also works to explain Ampere's laws.

I think I am getting it now, thank you.
 
Those ideas are subtle and require some attention and patience when your firstencounter them. Try looking for other interpretations and explanations and reread the ones you like several times over a few days...took awhile to for those ideas to sink in for me...

I'm always amazed at the people who first dreamed these kinds of things up...pretty classy!
 

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