Magnetic field within holey wire

[doive]

magnetic field within "holey" wire

Homework Statement

You have a wire running along the z axis with radius a, a hole is drilled in this wire, off-centre, but also in the z-direction. The centre of the hole is distance d from the centre of the wire, it has radius b. There is also uniform current density in the wire, J.

I need to find the magnetic field, B at the centre of the wire

Homework Equations

$$\oint$$B.dl=$$\mu$$₀$$I$$

The Attempt at a Solution

What I did initially is to split the wire into 3 sections. An inner cylinder, with nothing cut out of it, an middle ring which has the 'hole' drilled out of it, and an outer ring which has nothing cut out.

Since the inner cylinder and outer ring are entirely symmetric about the centre of the wire they contribute no magnetic field. (correct?)

you are left with a ring with a circular hole in it: http://s713.photobucket.com/albums/ww140/doive_photo/?action=view&current=wire.jpg
(image seems to be broken so here's a link)

now i can work out the area of that easily so can find the enclosed current, but I'm stuck at what to do from this point? I just can't seem to come up with any sensible expression with that line integral?

EDIT: I'm also quite new to tex... so apologies for the dodgy formatting

 

[RoyalCat]

Circular symmetry =\= no field contribution.

Try thinking about this another way, do you remember those gravity and electrostatics problems where you had a sphere with a hole cut through it? We used the principle of superposition to claim that there were equal negative and positive masses/charges where the holes were, the same concept applies here!

A hole, for all intents and purposes, is defined as an area where the current density is 0. We don't care if there's any actual material there.

$$0=\vec J + -\vec J$$

Use this breakdown to describe the hole and the whole cylinder and the solution should present itself! The result is quite astounding!

 

[doive]

Ah i see!

so instead of working out the B field for the quite complicated shape i just calculate B for the solid wire and then imagine a cylinder in the position of the drill hole with current flowing in the opposite direction.

So if i have B due to solid wire all i have to find is the B due to imaginary cylinder, which is just a simple "find B field a distance x from a infinitely long wire"
yes?

:D thanks very much for your swift help