where is the magnetic field-strength at radius . According to Ampère's circuital law, this line integral is equal to times the total current enclosed by the loop. The total current is clearly , since the loop lies outside the wire. Thus,

giving

for . This is exactly the same field distribution as that generated by an infinitely thin wire carrying the current . Thus, we conclude that the magnetic field generated outside a cylindrically symmetric -directed current distribution is the same as if all of the current were concentrated at the centre of the distribution. Let us now apply Ampère's circuital law to a circular loop which is of radius . The line integral of the magnetic field around this loop is simply . However, the current enclosed by the loop is equal to times the ratio of the area of the loop to the cross-sectional area of the wire (since the current is evenly distributed throughout the wire). Thus, Ampère's law yields

which gives

Clearly, the field inside the wire increases linearly with increasing distance from the centre of the wire.

If the current flows along the outside of the wire then the magnetic field distribution exterior to the wire is exactly the same as that described above. However, there is no field inside the wire. This follows immediately from Ampère's circuital law because the current enclosed by a circular loop whose radius is less than the radius of the wire is clearly zero.