Something wrong with Ampere's Law?

1. Oct 1, 2015

greswd

We use ∇ x B = μ0 J

Imagine a thin metal wire. We measure the curl at some distance from the wire and from the Biot-Savart law we know that it is not zero.

However, as this point is at some distance from the wire, the current density at that point is definitely zero.

I'm confused as to why this is the case.

2. Oct 1, 2015

andrewkirk

The J is not the current density at the point P where the curl is measured, but the current flowing through the area enclosed by the flow line obtained by following the magnetic field around from P until it gets back to P.

3. Oct 1, 2015

greswd

So from this area and the total current flowing through it, how do we put it in vector form? because J is a vector.

4. Oct 1, 2015

andrewkirk

Good point. I was thinking of the integral form of Ampere's Law. I don't think what I wrote makes sense in the differential form, which is what you're looking at.

Taking a step back, what version of the Biot-Savart Law are you using? From my understanding, the Biot-Savart gives a formula for B, which would be expected to be nonzero at a finite distance from an infinitely long, infinitely thin wire carrying a steady DC current. But it says nothing about the curl of B.

5. Oct 1, 2015

greswd

From the B-S law, we can construct the B vector field around an infinitely long wire. Then apply the curl differentials.

6. Oct 1, 2015

andrewkirk

I just did a quick check, and the curl of the mag field is indeed zero outside the wire, which matches the current density. M. Ampere will be relieved.

7. Oct 1, 2015

greswd

oh, something wrong with my calculations then. have you ever seen a derivation of the differential form of Ampere's Law by directly using the Biot Savart Law?

8. Oct 1, 2015

andrewkirk

Alas, my text just present's Ampere's Law as a fait accompli. Looking at the chronology, I suspect that Ampere's Law was discovered after the Biot-Savart and hence may have been derived from it. My text starts with the integral form of Ampere's Law and then derives the differential form using Stokes' Theorem.

9. Oct 2, 2015

greswd

idk if I'm right, but it appears to me that (via Stokes theorem), if the differential form is true then the integral form is definitely true, but if the integral form is true, it doesn't mean that the differential form is true.

10. Oct 2, 2015

ehild

We measure B, not curl B, at some distance of the wire.
Curl of a vector field is defined at a certain point as the limit of the line integral along a closed curve surrounding that point, divided by the area enclosed by the curve, when the area goes to zero. http://mathworld.wolfram.com/Curl.html
If the line integral of B happens to be equal to μ0 times the enclosed current, it does not necessarily mean that the limit (curl) exist at the point in question.
If you determine curl B outside the wire you get zero.

Last edited: Oct 2, 2015