Can you create this electric field?

On 2011-Mar-30, in physics, by paul

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 \vec{\nabla}\cdot \vec{v} means geometrically. The standard picture is that a vector field \vec{v} 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:

This vector field has a positive divergence, and no, this is not the field I'm asking you to create.

In case you’re interested, this field is simply given by \vec{v} = x\hat{x} + y\hat{y}. So this is a nice picture to keep in mind, and it helps you see why the vector field \vec{v} = x\hat{x} - y\hat{y} has zero divergence:

This field has zero divergence. Can you see why?

 

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:

 

Can you create an electric field like this?

 

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 \vec{v} = (1/x) \hat{x} 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:

    \[\vec{\nabla} \cdot \vec{v} = \frac{\partial v}{\partial x} = -\frac{1}{x^2}\]

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?