Understanding Divergence: Unit Vectors & Magnitude

[salman213]

1. I was just trying to understand what divergence means so I hope someone can help me out.

Well from what I have read if I take a vector field and use an infinitesimal region, if the vector going in is smaller than the vector going out there is positive divergence.

Does this mean if i make a circle with UNIT VECTORS there is ZERO divergence. Because `what is going in`, is the same as going out,

[URL][PLAIN]http://img11.imageshack.us/img11/9028/31455816et8.jpg


If they were NOT unit vectors or vector of the same magnitude then there Would be divergence? is that the correct concept?

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Hi salman213!

If you mean unit vectors all going out from a particular point, and a small circle round that point, then there is divergence, because all the vectors are going out of the circle.

Zero divergence means conservation of whatever-it-is …

often you have zero divergence everywhere except at "sources" and "sinks" …

eg an electric field with zero divergence except at the "singularities" where point charges are.

 

[nlightner]

Yes, you are correct. Divergence refers to the rate at which a vector field is expanding or contracting at a given point. If the vectors within an infinitesimal region are getting larger, then there is positive divergence. If they are getting smaller, then there is negative divergence.

In the case of a circle made with unit vectors, the vectors are all the same magnitude and direction, so there is no change in size or direction as you move around the circle. Therefore, there is zero divergence.

However, if the vectors were not unit vectors or if they had different magnitudes, then there would be a change in size or direction as you move around the circle, resulting in some level of divergence.

It's important to note that divergence can also be influenced by the shape of the vector field and the direction of the vectors. So while a circle made with unit vectors may have zero divergence, a different shape or direction of vectors could result in non-zero divergence.