The total current linking the closed path C
Line integral along the closed path C

Direction of Integration?

It is not necessary to know the direction of B before integrating!  Follow the simple rule illustrated below.

Curl your right-hand fingers around the closed path (Amperian loop), with them pointing in the direction of integration. A current passing through the loop in the general direction of your outstretched thumb is assigned a plus sign, and a current in the opposite direction is assigned a minus sign.

Example

Consider the magnetic field around an infinitely long wire due to a current I flowing through the wire.

But B has the same value a distance a away from the rod, hence

For the simple example, only one current linking the closed path C, therefore N = 1

This is the expression for the magnetic field due to an infinitely long wire.

Recall from Lecture #14

This is the magnetic field at point P due to the length of the wire AB

For an infinitely long wire, and , so that

This confirms the expression derived using Ampere’s law.

MAGNETIC FIELD INSIDE A WIRE

Let =   Current density (Am-2) Assuming the current is uniformly distributed through the wire

From Ampere’s Law

We can now plot the variation of the magnetic field strength inside and outside the wire.

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