Understanding the divergence theorem

1. polaris90

45
I'm having some trouble understanding what divergence of a vector field is in my "Fields and Waves" course. Divergence is defined as $divE=∇E = (∂Ex/∂x) + (∂Ey/∂y) + (∂Ez/∂z)$. As far as I understand this gives the strength of vector E at the point(x,y,z).
Divergence theorem is defined as $∫∇Eds$, where ds represents the area or volume of the vector field. In other words, I understand it as the overall strength of the vector field at a group of points composing a volume defined by the integral.
Could someone verify this for me?

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3. polaris90

45
At least pointing out what's wrong would help.

4. Ryoko

102
Divergence is not the strength of the field at a particular point. Divergence is the total amount of flux going into or leaving a volume at that point.

5. polaris90

45
Thanks, so would it be correct to say that divergence theorem refers to the amount of flux passing through a surface or volume?

6. Ryoko

102
Almost. Not through the volume, but rather being sourced or swallowed by the volume. For example, you cited ∇E which is often seen as part of one of Maxwell's equations. In this case, ∇E = q, which says the flux leaving a volume (the divergence) is equal to the amount of charge q contained in the volume. The charge q is sourcing e-field flux in this example.

7. polaris90

45
thank you all, that was very helpful

8. shawrix

18
Divergence theorem just makes it very clear.
What is flux?
Product of normal component of vector field and the surface element.

What is divergence of vector field?
Net flux diverging per unit volume.

What is Divergence?
Positive or negative divergence of a vector field at a point indicates whether the lines of force are diverging or converging at that given point.

By divergence theorem... The flux diverging from a given volume will be equal to the flux passing through the closed surface enclosing the volume.

For your question, the flux as explained above will be 'passing' through the closed surface enclosing the volume and the volume will be acting as the source of flux.

Consider a charge kept at the center of the sphere. The charge enclosed by this volume will be equal to electric flux lines passing normal to the closed surface. So the volume here is the source of flux whereas the flux is passing through the surface of the sphere.

Simple :]

Correct me if i am wrong.

Last edited: Feb 17, 2014