- #1

Telemachus

- 835

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- Homework Statement
- This doubt arised a few days, as a teaching assistant. A student asked about ampere's law in the context of this specific problem.

I had presented this problem: Calculate the magnetic field for a point localized at a distance "r" from the midpoint of a straight wire of length "2L" on which a current "i" is circulating.

So, the wire is along, let's say the z axis, and goes from z=-L to z=L.

- Relevant Equations
- We have Ampere's law in integral form: ##\oint \vec{B}\cdot d\vec{l}=\mu_0 i_{enc}##.

And the Biot Savart law: ##\vec{B}=\frac{\mu_0 i}{4\pi}\int \frac{d\vec{s}\times \hat{r}}{r^2}##

Using Biot-Savart law, it is easy to obtain in cylindrical coordinates, with ##\vec{r}=(r,\phi,z)## that

##\vec{B}(\vec{r})=\frac{\mu_0 i L}{2\pi r \sqrt{L^2+r^2}} \hat \phi##.

Then the question, for which I couldn't give a satisfactory answer, was that why can't we apply Ampere's law at the point z=0 in this problem? over that plane, the field is symmetric, and goes only along the ##\hat \phi## direction, so, in principle one could make a loop and obtain the field. But if we do this, the field obtained is the one for an infinite wire.

I tried to think why Ampere's law seems to fail in this case. For me it was clear that there is no symmetry in the z direction, there is no translational symmetry because of the finiteness of the wire. On the other hand, I know that Ampere's law is independent of the loop we take. This also poses some sort of contradiction, because if instead of a circle in the plane z=0 we use a different loop which doesn't live in the plane z=0, B is not constant anymore outside the plane. Something similar happens if we think of the surface which is enclosed by this loop. By Stokes theorem, one can deform that surface, until the point outside of the wire, and no current would be flowing through this "Stokes" surface, so there will be no current, and again a contradiction.

And then, the finite wire itself with a current that begins at a point, and ends at another point doesn't seem very physical neither.

So, I was looking for some discussion on this physical problem, formally, how to explain why Ampere's law can't be applied? is Ampere's law violated in this case? and why?

Thanks in advance.

And then, the finite wire itself with a current that begins at a point, and ends at another point doesn't seem very physical neither.

So, I was looking for some discussion on this physical problem, formally, how to explain why Ampere's law can't be applied? is Ampere's law violated in this case? and why?

Thanks in advance.