What is the physical significance of the divergence?

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Discussion Overview

The discussion centers around the physical significance of divergence in vector fields, particularly in the context of electromagnetics. Participants explore the conceptual understanding of divergence as it relates to sources and sinks within a vector field, as well as its implications in physical scenarios like electric fields and fluid flow.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant recalls that divergence is calculated as the del dot of a vector field and seeks clarification on its meaning in terms of sources and sinks at a point.
  • Another participant explains that divergence indicates the net movement of field lines away from a point, with positive divergence suggesting more lines originate than terminate at that point.
  • This same participant notes that a zero divergence indicates a balance of field lines originating and terminating, relating this to the absence of magnetic monopoles in electrodynamics.
  • Further, the concept of curl is introduced as a measure of rotation in the vector field, contrasting it with divergence.
  • A participant expresses appreciation for the explanations and reflects on the motivational potential of applying these concepts to electromagnetics.
  • Another participant suggests that understanding divergence should involve grasping its limit in terms of integrals, referencing external material for further exploration.

Areas of Agreement / Disagreement

Participants generally agree on the basic interpretations of divergence and its relationship to physical fields, but there is no consensus on the deeper implications or the best methods for understanding it. Some participants express uncertainty about the specifics of how divergence relates to sources and sinks at individual points.

Contextual Notes

Some discussions involve assumptions about the nature of vector fields and the interpretation of divergence, which may depend on the specific physical context being considered. There are references to mathematical definitions that may not be fully resolved in the discussion.

seang
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Hello;

I remember the days of muti variable calculus. The man said that divergence is equal to del dot the vector field. So on the exam he gave us a vector field, and I did del dot the given vector field and won big time.

The other day I decided my concentration would be electromagnetics. Now I need to know what divergence means. I understand that divergence gives you the scalar value of the source or sink at a point. Right?

It seems weird to me. That you decide the scalar value of a source or a sink at a POINT by considering the WHOLE vector field. I think I need help clearing this up.

For example, let's say I'm given a vector field A. Let's say del dot A = something. Does this mean that the vector field has a source equal to that something? At what point exactly? Is there only one source or sink?

This is the best that I can explain my troubles. I hope someone can help me. Thank you.
 
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If you consider a vector field as a physical field (for example, an electric field, or a flow of water), basically the divergence tells you how many field lines move away from the point (net). If the divergence is zero, as many lines originate and terminate at the point (this is for example, the reason that div B = 0 in electrodynamics: if it weren't, magnetic field lines could start somewhere which would prove the existence of magnetic monopoles). When the divergence is positive, more lines start at a point than terminate; basically: the greater the divergence, the more lines will start at the point. In other words: if I view the vector field as describing the flow of water, and I drop a ball near a point, the divergence will tell me if the ball will flow away from or towards the point.

The curl on the other hand (del cross) says something about the rotation: if I drop the same ball in the water, in how far will it flow around the point? The curl will be maximal if the water flows in a circle around my point, it will be zero if it flows radially away from it, for example (in the first case the divergence would be zero, in the latter it would be maximal).

PS Note that divergence and rotation are actually mathematically defined objects, and that this intuitive explanation can help you understand them. But even though they are called divergence and rotation for a reason, your intuition can deceive you.
 
that was really helpful, thanks. Between you and a TA friend of mine I think I've got this mostly figured out.

This stuff is really so interesting, when you apply it to something like EM. I can't believe they don't show multi variable students something like this. It would be highly motivational I feel.
 

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