proof that in general curl of B field is uJ

[Tor Fredrik]

 

Above they derive that curl of B is uJ. I know they use the identity

So since as underlined above that the derivatives of J is 0 we have that

But my problem is why is the derivative of J 0 in general. I have looked at a derivation for this:

  

And the end of this derivation is the following

But in the derivation they use that acceleration is constant and that it is a function of E showed in the orange box above. But E does not have to be constant since it is a function of r? So how is this a general derivation for the fact that divergence of current density is 0? 

[swansont]

One of the conditions of the derivation is that the current in the wire is constant.  

 

[Tor Fredrik]

58 minutes ago, swansont said:

One of the conditions of the derivation is that the current in the wire is constant.  

 

Why would that lead to that the divergence of the current density is 0. Can you show it mathematically?

Edited December 5, 2019 by Tor Fredrik

[swansont]

1 hour ago, Tor Fredrik said:

Why would that lead to that the divergence of the current density is 0. Can you show it mathematically?

The derivative of any constant is zero. That's such a basic understanding I'm not surprised it's not explicitly written down.

 

dC/dx = 0 for C = a constant