I’m teaching E&M this semester, and the following question arose as I was preparing a lecture on the mathematics needed for studying electricity and magnetism. When one talks about the divergence of a vector field, it’s nice to give a physical picture of what means geometrically. The standard picture is that a vector field has a non-zero divergence if the vector field either emanates from (positive divergence) or converges (negative divergence) to a point. Like this positive divergence field:

In case you’re interested, this field is simply given by . So this is a nice picture to keep in mind, and it helps you see why the vector field has zero divergence:

Ok. Now you have a geometrical picture of divergence, right? Use this understanding to tell me whether the field below has a non-zero divergence:

In this case, the geometrical picture I painted for you doesn’t give the right answer. This vector field does not diverge from a point, as the field lines are all precisely parallel. However although, the field does not emanate from a point, it is always pointing away from the y-axis. Furthermore, the algebraic formulation of the divergence of this vector field is easily calculated:

and this divergence is always negative, at any value of x.

So my question is this: is it possible to create an electric field that has globally parallel electric field lines yet still possessing a non-zero divergence just like the plot above? If so, what charge distribution would give rise to such a field?

Does my geometrical picture of divergence need to be modified?

My geometrical picture of the divergence doesn’t involve divergence from a point, but rather directly invokes Gauss’ theorem, meaning that I draw a little volume (or area in 2D) in my head, and try to see if there is a net flux in or out.

This gets around the issue you describe, but means you need to be able to do flux integrals in your head, which can be a bit tricky…

i want to understand more of this…i dont get the concept exactly how did the second vector field has 0 divergence?????

Consider a square centered on the origin; now consider the vector field’s flux

through this surface—I think you can see that there is no net flux through this surface.

Thus, there is no divergence—recall gauss’s law!

Does that help?

Could a zero divergence look like parallel lines that represent the vector of varying lengths that were not vertically parallel? i.e. one vector might be y=0 to y=5, x=0 and another might be y=1 to y=4, x = 1 ? Basically some parallel pairs of vectors would “end” (at some infinitesimal point) and other parallel pairs would end at another?

Marshall,

I’m not exactly sure of the picture you’re trying to paint in your question.

I was thinking about the field example I gave in the third figure; notice that if I took a

square shape (with sides parallel to the x & y axes), there would be a net flux into the box,

and therefore

and hence the field has a divergence (a negative divergence at that, since the vector field is larger at the left end of the

box’s side, and is inward-directed). Maybe thinking about Gauss’ Law is is the only way to properly think about divergence.