Ampere's Law and Biot-Savart Law confusion.

To avoid making this a TL;DR I'll have to be brief but I can expand if it's unclear.
I'm confused with magnetic field concepts, particularly Ampere's Law. It seems to suggest that the magnetic field around an arbitrary surface depends only on the current flowing through that surface; for example, the magnetic field within a current-carrying wire apparently depends only on the current within smaller radius 'r', where 'R' is the actual radius of the wire. I don't understand why this is the case?... though I suspect I'll need to study at a higher level to properly understand.
I wouldn't have such an issue with this if it wasn't that the magnetic field at the centre of, and along the central axis of, a current loop has a definite magnitude and direction. Surely the same problem exists? You could create a closed spherical surface within the loop, like a football with a hoop around it, and since the current though that surface would be zero, there would be no field according to Ampere's Law?
I've probably misinterpreted Ampere's Law... but since many of the problems I'm now asked to address feature complications, like a wire inside a cylindrical tube of current, I'm getting a bit worried. Any help much appreciated!
Thank you for reading.

Firstly, to avoid any confusion, stay clear of things like capacitors with amperes law, ampears law isn't completely general, and is only 1 part of maxwells 4th equation. Thats just an aside though.

Lets look at the mathematical form of Amperes law:



See a larger version here: http://upload.wikimedia.org/math/4/1/2/4125ace626e2bd298c5976f94077a660.png

I'd avoid the 2nd term in this equation, it isn't necessary for you atm and may just be confusing. Look at the one on the left and right.

Lets look at the left bit first. In particular note the circle on the integral, this represents a complete closed path. In other words you are integrating round a closed curve. B is the magnetic field vector with a strength and direction, and dl is a infinitesimal vector pointing in the direction that you are doing the integral around. In other words, the term on the left takes all the magnetic field vectors, and extracts their component in the direction of the arbitrary path and then adds them together. Since we have a component, note the result of this integral will be a number, which is exactly what you would expect!

Now look at the result on the right, this is the number that you end up with. It has a constant, which is the permitivity of free space. And the current. The subscript enc in this case means the enclosed current. So if you have a wire that passes through your arbitary path, then the current passing through the wire will be the same number you get by the integration.

Note: in 80% of cases you won't need to do an actual integration, because the geometry is simple enough to just do a sum of dot products, or even just some multiplications.


The key point to understand is that the path you choose (represented by C on the integral) is completely arbitrary, the left side is equal to the right side regardless of C, provided that C is closed! So you could have a path that doesn't enclose the wire, but in that case the magnetic field vectors would end up cancelling and you would get 0=0.

Thank you for the response.
I actually understand the definition of Ampere's Law. Your last sentence is the bit that was confusing me before. Why do the magnetic field vectors end up cancelling if there's no current running through them? Also, is C a closed loop or a closed surface? And if I took a path C within a loop of current, since there's no current running through the path C, shouldn't there be no magnetic field? And yet this doesn't appear to be the case.