Determining charges and current distributions

[TubbaBlubba]

Homework Statement

I need to solve a bit of a simple "reverse" problem that I'm unable to find treated in any detail, probably because it's actually reasonably straightforward. What I need to do is to "compute the charge and current distribution that give rise to [the following] field."

Homework Equations

F[/B] = 2ye_y when r < 1
F = 0 when r > 1.

The Attempt at a Solution

Obviously, this gives us a potential function φ = -y² (for r < 1).

Now, we want to determine all the point charges, line charges, surface charges, volume charges, line currents, surface currents, and volume currents. Surface charges and currents, and volume charges and currents, are simple - they can just be computed from the field using the dot and cross product of the normal with the field difference, and the divergence and curl of the field respectively. I won't be typing them all out, but .e.g for the volume charge ρ we have

∇⋅F = 2 = ρ for r < 1,
(and ρ = 0 for r > 1.)

But when it comes to point charges, and line charges/currents, I'm just told that, e.g. for a point charge q we have

φ = q(4πr)^-1.

Um, so presumably we don't have q = -y²4πr, because that's, well, nonsense in this context.
Instead, I reason that there is no part of my potential of this form, and therefore we have no point charge; q = 0 (which I know is the correct answer; it is also clear from the form of the field). What I wonder is the following: is this reasoning adequate, or is there some handy calculation (presumably some integral using the Dirac delta function) I could go through to show that there is indeed no point charge (and similarly for the other two)?

Thank you in advance; I hope this isn't too general or diffuse a question.

 

[TubbaBlubba]

OK, I think I figured it out after a long while of pondering including four different textbooks and examining multiple previous exams. When they ask me to find point charges, if any, they want me to look at the form of the integral and maybe check for a charge at the origin with a limiting volume integral as r goes to zero. Similarly for line charges and currents if the field has a "cylindrical" component. Finally, for surface charges/currents, the surface in question is the one where the field is not described (in this case, sphere where r = 1).

You know, when they give us assignments calling for methods not remotely covered in the course literature (and EM Field Theory is not until next half-term), it'd be pretty nice if they could be a bit more descriptive of what they're looking for. That's mathematical physics for you, I guess...

(By the way, I'm aware that I'm emitting epsilon-naught above. As do the assignments in this course. Yeah.)