Amperes law: magnetic field inside an infinite slab containing a volume current.

[JFuld]

Homework Statement

an infinite slab is centered on the xy plane. the top of the slap is at z=a/2, the bottom of the slab is at z= -a/2. A volume current (J) is set up within the slab. find B everywhere

J = k (1 - z^2/a^2) \hat{y}

Homework Equations

amperes law

The Attempt at a Solution

the only thing I am confused about is finding B inside the slab.

I was going to set up my amperian loop as a rectangle, which is oriented perpendicular to the volume current.

The rectangle would extend a distance z above and below z=0 (z still inside the slab)

however my textbook mentions B is discontinious at a surface current, so i am uncertain if my amperian loop takes that into account.

But given the stuff we have learned so far, I am very confident that we are supposed to apply amperes law to this situation.

Can anyone explain if you can still apply amperes law for points inside a volume current?

 

[rude man]

The only way i see is to
(1) detrmine the vector potential A = (1/4π)∫idv/r
where i = current density across a surface in elemental volume dv anywhere within the slab,
and r = distance from dv to the point of observation. Since i has a y component only, so will A. So A = A_y y.

dv = dx*dy*dz and extends from -∞ < x < ∞, -∞ < y < ∞, and -a/2 < z < a/2
for the part inside the slab. For the outside, integration is limited to |z| > a/2.
r is the distance from each dv to (x,y,z).

This looks like a horrible pair of volume integrals to me!

(2) If you did solve for A_y you could then find B from
B = μ*curl (A_y y).

Sorry this is the best I could do.

 

[JFuld]

so I think I may have left out the most important piece of information to this problem:

the slab is made of a diamagnetic material (linear media, magnetization antiparallel to B)

then inside the slab, you apply amperes law for the auxilary field H

B=μ_{o}(1+χ_{m})H

where the magnetization M =χ_{m}H

So this discontinuity I was so worried about is due to the magnetization?
(M should produce a surface current, which is exactly the source of B's discontinuity)I am pretty comfortable doing these types of problems mathematically, but is my reasoning sound?