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The Ampere circuital theorem can be expressed as $$\oint_lB\operatorname dl=\mu_0\sum I.$$ This does not seem to say anything about the wire which carries the current. If the wire has shape, for example a prism or a cylinder with varying radius, how to use the theorem?

An example problem. The cross sectional surface of an infinitely long straight wire is in such a shape,

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How to find the magnetic flux density around it?

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I assume that there is a uniform current density in the wire, and no current in the black circle. This is a famous problem because it is solved so simply. Treat it as two wires, one wire being the big circle with uniform current density, and the second wire with the same current density in the opposite direction in the black circle. Then just add vectorially the field from each wire.

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