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Question 5.15: A thick slab extending from z = −a to z = +a (and infinite i...

A thick slab extending from z = −a to z = +a (and infinite in the x and y directions) carries a uniform volume current J =J \hat{ x } (Fig. 5.41). Find the magnetic field, as a function of z, both inside and outside the slab.

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By the right-hand-rule, the field points in the -\hat{ y } direction for z > 0, and in the +\hat{ y } direction for z < 0. At z = 0, B = 0. Use the amperian loop shown:

\oint B \cdot d l =B l=\mu_{0} I_{ enc }=\mu_{0} l z J \Rightarrow B =-\mu_{0} J z \hat{ y } \quad(-a<z<a) . \text { If } z>a, I_{ enc }=\mu_{0} l a J ,

so

B =\left\{\begin{array}{l}-\mu_{0} J a \hat{ y }, \text { for } z>+a; \\+\mu_{0} J a \hat{ y }, \text { for } z>-a.\end{array}\right\}
5.15

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