The current density inside a long, solid, cylindrical wire of radius a = 5.0 mm is in the direction of the central axis and its magnitude varies linearly with radial distance r from the axis according to j = j0r/a, where j0 = 420 a/m2. find the magnitude of the magnetic field at a distance (a) r=0, (b) r = 2.9 mm and (c) r=5.0 mm from the center.

Explanation:

Given that,

Radius a= 5.0 mm

Radial distance r= 0, 2.9 mm, 5.0 mm

Current density at the center of the wire is given by

Given relation between current density and radial distance

We know that,

When the current passing through the wire changes with radial distance,

then the magnetic field is induced in the wire.

The induced magnetic field is

...(I)

We need to calculate the induced current

Using formula of induced current

We need to calculate the magnetic field

Put the value of induced current in equation (I)

(a). The  magnetic field at a distance r = 0

Put the value into the formula

The  magnetic field at a distance 0 is zero.

(b). The  magnetic field at a distance r = 2.9 mm

The  magnetic field at a distance 2.9 mm is

(c). The  magnetic field at a distance r = 5.0 mm

The  magnetic field at a distance 5.0 mm is

Hence, This is the required solution.