Consider a thick SLAB of current. The slab is infinite in (both) x and y, but fi

Question

Consider a thick SLAB of current. The slab is infinite in (both) x and y, but finite in z. So we must think about the volume current density J, rather than 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 J depends on height linearly, i.e. J = J_0 abs(z) x^, inside the slab (but is 0 above or below the slab), where abs(z) is the absolute value of z. Find the B field (magnitude and direction) everywhere in space (above, below, and also, most interestingly, inside the slab!)

Explanation / Answer

Here we will be applying amperes law just like we used to do with Gauss law,

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...... inside the slab

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

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