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Or equivalently does Ampere's law (point form) says that at a POINT where there's no current flow, there's no magnetic field at all? Are the points (except those containing the wire) within an Amperian loop with a single current carrying wire in it such points? Since current density at those points is zero as the current is flowing only through a single strand of wire inside the loop,by Ampere's law the curl should be zero at those points. But there should be a non-zero magnetic field present at those points due to the current carrying wire.

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Can a point in space exist such that the magnetic field there is non-zero but the curl of the field is zero?

This is an interesting question. One would think that since $\nabla \cdot \textbf{B}=0$ at all points, if $\nabla\times\textbf{B}=0$ at a point, maybe $\textbf{B}$ has to be zero at that point, after all, the curl and divergence of a field are supposed to completely determine the field, right? (Otherwise, how would Maxwell's equations determine the electric and magnetic fields--all they do is specify the curl and the divergence of the two fields!)

While this impulse is in the right direction, it is not exactly correct. The correct statement is that if the divergence and curl of a field are specified at all points in space then one can deduce the field at every point in space. More explicitly, the Helmholtz theorem states that $$\textbf{F}(\textbf{r})=-\nabla\bigg(\frac{1}{4\pi}\int\frac{\nabla\cdot\textbf{F}(\textbf{r}')}{|\textbf{r}'-\textbf{r}|}d^3\textbf{r}'\bigg)+\nabla\times\bigg(\frac{1}{4\pi}\int\frac{\nabla\times\textbf{F}(\textbf{r}')}{|\textbf{r}'-\textbf{r}|}d^3\textbf{r}'\bigg)$$ As you can see, the curls and the divergences would need to decay fast enough for these integrals to converge. This simply translates to the fact that we don't expect the sources (such as charge or current distributions) to extend indefinitely in space.

So, as you can see, the field itself at a point gets contributions from the divergences and curls of the field at all points in space$-$not just at the point where we are calculating the field. In particular, for the magnetic field, since the divergence is always vanishing, we can write $$\textbf{B}(\textbf{r})=\nabla\times\bigg(\frac{1}{4\pi}\int\frac{\nabla\times\textbf{B}(\textbf{r}')}{|\textbf{r}'-\textbf{r}|}d^3\textbf{r}'\bigg)$$ So, as you can see, the vanishing of the curl of the magnetic field at a point doesn't ensure that the field itself would be zero at that point. If the curl of the magnetic field is zero everywhere (i.e., there are no currents at all) then the field would be zero at all points, of course.


As you rightly observe, if it were the case that the field would be zero at all points where the curl is zero then an Amperian loop around a wire would have zero magnetic field at all of its points and it would calculate a vanishing circulation around the wire, in contradiction with the Ampere's rule. But thankfully, we can be sure that no such contradiction is implied or exists.

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  • $\begingroup$ Thank you for helping me find a very helpful theorem... $\endgroup$ – user218328 May 5 '20 at 16:14
  • $\begingroup$ So as per the Helmholtz theorem, divergence at a point too should behave the same in order to satisfy Gauss' s law right? $\endgroup$ – user218328 May 5 '20 at 16:17
  • $\begingroup$ @user218328 Glad that you found it helpful :) I am not sure what you mean. But if you mean that the electric field at a point need not be zero just because the divergence of the electric field is zero then yes. If the divergence is zero at a point, it just means that there is no charge at that point. But as long as there are charges elsewhere, there can absolutely be an electric field at a point. $\endgroup$ – Dvij D.C. May 5 '20 at 16:21
  • $\begingroup$ Thank you again for your clarifications $\endgroup$ – user218328 May 5 '20 at 16:48
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Can a point in space exist such that the magnetic field there is non-zero but the curl of the field is zero?

Yes. For example, take an infinite current carrying wire. The curl at any point which is not on the wire, will be zero. However, magnetic field at all those points would always be some non zero value.

Or equivalently does Ampere's law (point form) says that at a POINT where there's no current flow, there's no magnetic field at all?

Not at all. It just says that the curl of the field at that point will be zero.

All your following questions have already been addressed in the example given above.

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  • $\begingroup$ By Stokes theorem,the infinitely large sum of curls within a planar surface in a field is the closed line integral of the field through its boundary.So does it means that we are adding zeroes when the points are not on the wire inside the surface? I know this doesn't oppose your answer as there are points with non zero curl within the surface which may give the line integral through the boundary.I just wanted to clarify $\endgroup$ – user218328 May 5 '20 at 9:45
  • $\begingroup$ @user218328 Yes, except the curls which are at the wire's location. $\endgroup$ – user258881 May 5 '20 at 9:47
  • $\begingroup$ The same goes for divergence at a point right? $\endgroup$ – user218328 May 5 '20 at 16:19
  • $\begingroup$ @user218328 The divergence is zero everywhere, no matter where you measure it. $\endgroup$ – user258881 May 5 '20 at 16:21
  • $\begingroup$ Oops I meant it for electric field. Sorry for not being specific. $\endgroup$ – user218328 May 5 '20 at 16:43

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