Undergrad Ampere-Maxwell law seems to contradict causality?

[Bob44]

Let us take the Ampere-Maxwell law

$$\nabla \times \mathbf{B} = \mu_0\,\mathbf{J}+\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}.\tag{1}$$

Assume we produce a spark that is so fast that the $\partial \mathbf{E}/\partial t$ term in eqn.$(1)$ has not yet been produced by Faraday’s law of induction
$$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}\tag{2}$$
since the current density $\mathbf{J}$ has not yet had time to generate the magnetic field $\mathbf{B}$.

By integrating eqn.$(1)$ and using Stokes law we find

$$\oint \mathbf{B}\cdot d\mathbf{l}=\mu_0 I,\tag{3}$$
$$B=\frac{\mu_0 I}{2\pi r}.\tag{4}$$
This seems to imply that a tangential magnetic field with strength $B$ appears instantly around the spark at all distances $r$.

Does this contradict causality?

 

[Dale]

Assume we produce a spark that is so fast that the ∂E/∂t term in eqn.(1) has not yet been produced

That is an impossible assumption. The differential Maxwell’s equations hold at a point. So there is no delay

the current density J has not yet had time to generate the magnetic field B.

Similarly here.

 

[Bob44]

But the tangential magnetic field produced by the spark due to Stokes law is still instantaneous at all distances. This seems to contradict causality regardless of the radiation that later travels from the spark at the speed of light.

 

[Dale]

But the tangential magnetic field produced by the spark due to Stokes law is still instantaneous at all distances.

No. It just means that your assumption is wrong.

Look, your assumption is basically that something happened which is so fast that $\partial/\partial t$ terms were zero. But that is patently a bad assumption. When things happen fast is precisely when $\partial/\partial t$ terms are largest.

Your conclusions based on this wrong assumption are simply wrong. In fact, proof by contradiction is a common way of proving that an assumption is false.