Biot-Savart law and Ampere's law

[Nathanael]

If we integrate the magnetic field from the Biot-Savart law for an infinitely long straight wire, we can get $|B|=\frac{\mu_0 i}{2\pi R}$ with R being the shortest distance from the wire to the point in space.

If we use Ampere's law (with a circle of radius R centered on a wire with the normal of the circle parallel to the wire) then we can get the same relationship much more easily.

However, when we use Ampere's law we are not assuming anything about the length of the wire, it can be 1 cm as long as it goes through the circle.

But if we did not integrate the Biot-Savart law from -∞ to ∞ then we would not get this result. Yet Ampere's law implies that it's true for all wires regardless of length (with a restriction on the points in space for which it applies, of course).

What am I missing?

 

[jtbell]

However, when we use Ampere's law we are not assuming anything about the length of the wire,

Actually, when we apply Ampere's Law (alone) to a straight wire, we have to assume the wire is infinite in length (or at least "long enough" that it's "practically infinite") in both directions, in order to maintain the current. If the wire is only 1 cm long, what happens to the current at the ends of the wire?

Integrating the Biot-Savart law for a finite wire segment (i.e. not a closed loop, nor goes off to infinity at both ends) doesn't work, either.

By the way, you're in good company here. Maxwell had to overcome basically this problem in order to get a mathematically consistent set of equations for electrodynamics. The version of Ampere's Law that you're trying to use is incomplete. Look up (or recall, if you've studied this before) how Maxwell fixed Ampere's Law.

 

[Nathanael]

Actually, when we apply Ampere's Law (alone) to a straight wire, we have to assume the wire is infinite in length (or at least "long enough" that it's "practically infinite") in both directions, in order to maintain the current. If the wire is only 1 cm long, what happens to the current at the ends of the wire?

I just kind of vaguely imagined the current coming from and going in some kind of ambiguous 'charge reservoir.' I didn't think the details were important. (Hey, it worked for thermodynamics! )

Integrating the Biot-Savart law for a finite wire segment (i.e. not a closed loop, nor goes off to infinity at both ends) doesn't work, either.

Just for the same reason as above? If you have more to say, I'm curious.

edit:

The version of Ampere's Law that you're trying to use is incomplete. Look up (or recall, if you've studied this before) how Maxwell fixed Ampere's Law.

I haven't studied this yet but I will look it up.

 

[Nathanael]

I see (sort of):
If we had some kind of "charge reservoirs" then the transfer of charge between them would cause the electric field to vary in time which is unaccounted for in this form of ampere's law. I guess I will understand it better when I properly study electromagnetism. Thanks jtbell.