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Using Ampere's Law to Find the Magnetic Field Around a Long Straight Current Carrying Wire

Ampere's Law equates the line integral of the magnetic field around an arbitrary loop to %mu-0 *the current passing through the loop:The loop may be any closed loop. If no current passes through the loop or if it passes through and back without threading the loop, then it is deemed not to have passed though and the line integral is zero.

Many situations where Ampere's Law may be used possess symmetry which may greatly simplify the problem.

The long straight wire above carries a currentUsing the right hand grip rule gives us the sense of the magnetic field. Now we can use Ampere's Law to find the magnetic fieldwhich because of the straightness of the wire, possess circular symmetry.

Because of the circular symmetry of the problemhas the same magnitude at every perpendicular distance r from the wire, andis parallel toat every point around the loop drawn on the diagram above hence

Hence

The magnetic field at a point due to a long straight wire carrying a current depends only on the size of the current and the distance of the point from the wire.

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