Magnetostatics of Current-Carrying wire A question has been nagging at me about Faraday's Law as related to a wire with a constant current:
If you have a circular loop of wire with some small resistivity, connected to a battery so that it has a constant DC current, the electric field inside the wire is going to be longitudinal and proportional to the current density. In particular, is going to be something like $\vec{E}=E_0\hat{\phi}$, choosing cylindrical coordinates with the origin at the center of the loop. The curl of this field is obviously nonzero, so applying Faraday's Law there should be a linearly growing magnetic field in the Z direction, at every point inside the wire. But this seems like nonsense; we're supposed to be in a steady-state, and it doesn't seem physically plausible to me that there is a field that can be growing arbitrarily large as we keep our battery plugged in, or that its growth should be affected by the shape of the wire loop. So I expect I am missing some contribution that would cancel this effect. But what it is? I feel like I must be missing something obvious, but can't for the life of me see what it is.
 A: If you apply Faraday's law here, the net emf around the loop is zero (as expected). You forget that there is a battery in the loop, which has a large, opposite electric field across it. So $\varepsilon=\varepsilon_{loop}+\varepsilon_{battery}=0$. Which means that $-\frac{d\Phi_B}{dt}=0$, so $B=constant$, which means that the loop on its own will not induce any magnetic field, but it can survive in a system with a constant magnetic field without any change.
A: First off, the self-inductance of any closed loop like this inhibits current flow; it doesn't swell it infinitely. That's back-emf. Second, while the total magnetic field of the loop will be in the $Z$ direction at the origin, it won't be so in the wires. Remember that the magnetic field of any small section of current-carrying wire is a cylinder. The sum of all these little contributions is what gives us the total field that you described, which looks like this.
So, summing up, when the circuit is first connected, self-inductance in the wires themselves and in the loop as a whole prevent the current from reaching its final state instantaneously, and the resistance of the wire and the internal resistance of the battery cause the current to reach equilibrium at its steady state. At that point, the loop has a steady magnetic field, but doesn't have to worry about self-inductance anymore.
I may have misunderstood you question, but I think that will help.
