I Curl of current density

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The discussion revolves around the behavior of the current density vector, ##\vec J##, in a 2D physical problem, which is non-zero everywhere except at specific points. It is suggested that ##\vec J## decays to zero as it approaches the centers of vortices, where the current density is assumed to vanish. The implications for the magnetic field are also considered, with the assumption that it similarly decays to zero at the vortices due to the absence of current. The conversation seeks further clarification on the expressions and visual representations related to these phenomena. Overall, the relationship between current density and magnetic field behavior in the context of vortices is a central focus.
fluidistic
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Hello, I have derived an expression for ##\vec J## in a particular, unusual physical problem in 2D. The expression is different from ##\vec 0## everwhere in the material except at at least two different points.

Can I conclude that the only way for this to occur is that ##\vec J## itself decays to ##\vec 0## the closer it gets to the center of the vortices? I assume that the current density vanishes exactly at the center of vortices of currents.

Also, what can I conclude about the magnetic field?
 
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Would you be able to share the expression or perhaps have an image?

I would assume that the magnetic field decays to zero since at those vortices the current goes to zero.
 
Thread 'Gauss' law seems to imply instantaneous electric field'

Thread 'Gauss' law seems to imply instantaneous electric field'

Imagine a charged sphere at the origin connected through an open switch to a vertical grounded wire. We wish to find an expression for the horizontal component of the electric field at a distance ##\mathbf{r}## from the sphere as it discharges. By using the Lorenz gauge condition: $$\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}=0\tag{1}$$ we find the following retarded solutions to the Maxwell equations If we assume that...
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Thread 'Gauss' law seems to imply instantaneous electric field (version 2)'

Thread 'Gauss' law seems to imply instantaneous electric field (version 2)'

This argument is another version of my previous post. Imagine that we have two long vertical wires connected either side of a charged sphere. We connect the two wires to the charged sphere simultaneously so that it is discharged by equal and opposite currents. Using the Lorenz gauge ##\nabla\cdot\mathbf{A}+(1/c^2)\partial \phi/\partial t=0##, Maxwell's equations have the following retarded wave solutions in the scalar and vector potentials. Starting from Gauss's law $$\nabla \cdot...
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