Now the sheet of current has become a thick SLAB of current. The slab is infinit

Question

Now the sheet of current has become a thick SLAB of current. The slab is infinite in (both) x and y, but finite in f z. So we must think about the volume current density vector J, rather than vector K. The slab has thickness 2h (It runs from z = - h to z = +h). Let's assume that the current is still flowing in the +x direction, and is uniform in the x and y dimensions, but now vector J depends on height linearly, i.e. vector J = J_0|z| cap x inside the slab (but is 0 above and below the slab). Find vector B (magnitude and direction) everywhere in space (above, below, and also, most interesting, inside the slab).

Explanation / Answer

Here we will be using amperes law to calculate B, just like we used Gauss law to calculate E.

By amperes law,

integral Bdr = uo i enclosed

Lets take a amperes rectangle in yz plane symmetrical to both side of origin.

B*2y = 2uo* integral Jy dz

B*2y = 2uo yJo integral zdz

B = uoJo z^2/2....for |z| <h i.e. inside slab

B = uoJoh^2/2 for |z| > h i.e. above amd below the slab

Total B = uo Jo (h^2+z^2) /4   

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