A long wire carries a current uniformly distributed over a cross section of the wire. (a) We have to show that the magnetic energy of a length l stored within the wire equals . (b) We have to show that the inductance for a length l of wire associated with the flux inside the wire is .

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Let the radius of the wire be R. It is given that it is carrying a current . Therefore, the current density in the wire is

We use the Ampere’s law for finding the magnetic field within the wire at a distance r from its axis. The magnetic field will be cylindrical. Therefore,

Magnetic energy volume density is given by

Therefore, the energy density of the magnetic field at a distance r from the axis of the wire carrying current will be

The energy stored as magnetic field within a length l of the wire will therefore be given by the following integral:

The inductance of a length l of the wire can now be found from the relationship

Therefore,