A long wire has a circular cross-section with radius . The current density in the wire is where is the distance from the axis. Calculate the magnetic field at an arbitrary point inside or outside the wire.

A long wire has a circular cross-section with radius . The current density in the wire is where is the distance from t

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Problem: 

A long wire has a circular cross-section with radius . The current density in the wire is where is the distance from the axis. Calculate the magnetic field at an arbitrary point inside or outside the wire.

Solution:

For inside the wire:

The current enclosed by the Ampere's loop is given by 

or,

or,

The magnetic field inside the wire is given by 

or,

or, [ANS]

For outside the wire :

The current enclosed by the Ampere's loop is given by

or,

or,

The magnetic field outside the wire is given by 

or,

or, [ANS]

The magnetic field becomes maximum at , so

or,

or, [ANS]

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