Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 15D / a

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- Solution a

Solution

Translation symmetry in the plane and reflection symmetry across it force the magnetic field to be uniform on each side, parallel to

e_{y}

, and opposite on the two sides. Apply Ampère's law to a rectangular loop perpendicular to the sheet whose two long sides have length

L

. The enclosed current is

K L

, so

2 B L = μ_{0} K L .

The right-hand rule fixes the directions:

B (z > 0) = - \frac{μ _{0} K}{2} e_{y}, B (z < 0) = \frac{μ _{0} K}{2} e_{y} .

Consequently

e_{z} \times B (0^{+}) - e_{z} \times B (0^{-}) = \frac{μ _{0} K}{2} e_{x} - (- \frac{μ _{0} K}{2} e_{x}) = μ_{0} K,

which is the magnetic jump condition for a surface current density.

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