How much time for a magnetic field to disappear?

Question

Consider current flowing through a conductor. When I disconnect the power to the conductor there should be a surge in the magnetic field the moment the current touches zero. My question is how long that magnetic field lasts after ?

Answer 1

If you have a steady current through a long wire then you can get a nice static nonzero magnetic field and a nice static zero electric field.

Since the fields are static we know that:

$$\vec \nabla \times \vec E=-\frac{\partial B}{\partial t}=\vec 0$$ and

$$-\mu_0\vec J +\vec \nabla \times \vec B= \mu_0\epsilon_0\frac{\partial E}{\partial t}=\vec 0.$$

But once you start changing the current $\vec J$ then the current no longer cancels $\vec \nabla \times \vec B$ and so the electric field immediately starts to change right where the current is. And at first there is no actual change in the magnetic field.

But as the electric field starts to change, a field develops where $\vec \nabla \times \vec E$ is nonzero, and that makes the magnetic field start to change.

Now, this approach has both the electric and magnetic fields each exist, and you take their initial values and from them get how they change over time from Maxwell, but you have to know the current.

But if you want to see how electric and magnetic field could depend on charges and currents there a completely different method. Which can directly assign fields in a causal way based on charges and currents, one such example is Jefimenko's equations:

$$\vec E(\vec r,t)=\frac{1}{4\pi\epsilon_0}\int\left[\frac{\rho(\vec r',t_r)}{|\vec r -\vec r'|}+\frac{\partial \rho(\vec r',t_r)}{c\partial t}\right]\frac{\vec r -\vec r'}{|\vec r -\vec r'|^2} -\frac{1}{|\vec r-\vec r'|c}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\mathbb{d}^3\vec r'$$ and $$\vec B(\vec r,t)=\frac{\mu_0}{4\pi}\int\left[\frac{\vec J(\vec r',t_r)}{|\vec r -\vec r'|^3}+\frac{1}{|\vec r -\vec r'|^2}\frac{\partial \vec J(\vec r',t_r)}{c\partial t}\right]\times(\vec r -\vec r')\mathbb{d}^3\vec r'$$ where $t_r$ is actually a function of $\vec r'$, specifically $t_r=t-\frac{|\vec r-\vec r'|}{c}.$

These reduce to Coulomb and Biot-Savart only when those time derivatives are exactly zero, which is statics. So Jefimenko is an example of proper time dependent laws for the electromagnetic field. Note that both the electric and the magnetic part of the electromagnetic field have parts that depend on the time variation of current.

So if you use these equations, the change in current directly causes both electric and magnetic fields. But when the current changes at place-time $(\vec r_1,t_1)$, there is an electric and a magnetic field. But the field exists only at place-times $(\vec r_2,t_2)$ where $t_2=t_1+\frac{|\vec r_2-\vec r_1|}{c}$.

So of the current isn't zero yet, you could see a magnetic field due to the current and a magnetic field due to the change in current. But don't think the one due to the current is as simple as the static field. Basically you have to wait to see the changes and you see the changes from the part of the wire closest to you first, so you'll see the old strong current on all the wire except the part closest to you and see the magentic field from that, and then for the part closest to you you'll see less current (and hence less of that contribution to the total magnetic field) plus you will see a new source, the change in current.

Changes in currents are simply new sources of electric and magnetic field and they have that common source.

So when the current changes, there is a spherical shell from each bit and each shell expands at the speed $c$ and on that shell there is both a new electric and a new magnetic field.

If your current goes to zero and then stays there then both sources are zero and so there is an expanding shell from that moment and inside the region contained inside all the shells is something with no field. But an infinite wire has no such region, so the field is never imperfectly zero. There is always a part super far away where you are still seeing the contribution from what used to be the steady current (since stuff far away contributed based on what they were doing long ago).