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I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following:

enter image description here

E has an ambiguous divergence and I think I know why. Its strength seems to weaken in the $\hat r$ direction, and if $R(r) \propto \frac{1}{r}$, then if the vector field is depicting $\vec F$, then $$\hat r \cdot (\nabla \cdot \vec F) = \frac{\partial (r F_r)}{\partial r} \propto \frac{\partial (r \ \frac{1}{r})}{\partial r}$$ which is trivial unless $r=0$ in which it is undefined. So, we can assume the divergence is non-zero unless in this special form, which would cause it to be zero everywhere except for a singularity at $r=0$.

For C, however, I am not sure how to justify its ambiguous curl. My lecturer wrote that

It could be the case that the curl is zero everywhere except at the center, but it could also be the case that the curl is nonzero everywhere. If the magnitude were changing with distance like $1/s$, then any line integral around a closed loop will be zero unless it encloses the origin.

I suspect that these two things have something to with the motivation for Gauss's Law and Ampere's Law, as the singularities sound like whether there is the presence of a charge or current density. Anyway, I don't see how "If the magnitude were changing with distance like $1/s$, then any line integral around a closed loop will be zero unless it encloses the origin." justifies the ambiguity. I see how it does for divergence, but not for curl, as I don't think my justification works as the radial component doesn't work that way. Given if my justification for why E's divergence is ambiguous, why is C's curl ambiguous?

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  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$
    – user4552
    Dec 12, 2018 at 23:22

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Take a counterclockwise loop between $\theta_a, \theta_b$ and $r_a,r_b$ where $r_b>r_a$. The circulation is the magnetic field at constant $r_b$ minus the side at constant $r_a$ times their respective lengths (the constant $\theta$ sides are perpendicular to the flow so vanish).

$$\text{Circulation} = B(r_b)r_b(\theta_b-\theta_a)-B(r_a)r_a(\theta_b-\theta_a)$$

If $B(r)=B_0/r$ then the circulation and thus the curl vanishes (we are free to take a limit as the size of the loop goes to zero), like your lecturer said. Moreover, if for instance $B(r)=B_0/r^2$ then you have a negative curl everywhere except the origin.

Here is a example of the kind of loop I am talking about. Since $B$ is pointing in the $\theta$ direction the line integral over the left and right side vanishes, and over the top and bottom sides it is just the value of the $B$ field times the length of the curve. enter image description here

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  • $\begingroup$ I have trouble understanding your answer, I think maybe because I don't understand this magnetic field property and why it's circulation is defined the way it is here. Is there a more general example/thought experiment/justification without going into a physics recourse? $\endgroup$
    – sangstar
    Dec 13, 2018 at 15:18
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    $\begingroup$ There's nothing special about the magnetic field here. It is an arbitrary vector function that happens to only point in the $\theta$ direction like in your picture. The circulation is just the line integral around a closed loop, just like your lecturer says. Could you be more specific about which part you don't understand? $\endgroup$
    – octonion
    Dec 13, 2018 at 22:27
  • $\begingroup$ I added a picture to clarify $\endgroup$
    – octonion
    Dec 13, 2018 at 22:37

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