What is the physical significance of the divergence?

[seang]

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.

 

[CompuChip]

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.

 

[seang]

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.

 

[quasar987]

The physical interpretation of div comes from this characterizations in terms of the limit of an integral. Your goal should be to understand this limit.

http://en.wikipedia.org/wiki/Divergence#Physical_interpretation