Question

PROBLEM 02.07 A very long solenoid is filled with an inhomogeneous paramagnetic material whose magnetic susceptibility depends on the distance r from solenoid's axis as r a, where a is some positive constant. The value of the magnetic field on the axis is Bo Find the magnetization M(), the magnetic field B(), and all the bound currents everywhere.

Answer 1

(i) The magnetization, M(r) which will be given by -

M(r) = $\chi$(r) . H

M(r) = $\chi$(r) . (B₀ / _$\mu$0)

where, B₀ = magnetic field value on the axis

_$\mu$0 = permeability of free space

$\chi$(r) = magnetic susceptibility depends on distance 'r' from solenoid axis = a r²

then, we get

M(r) = (a r²) (B₀ / _$\mu$0)

(ii) The magnetic field, B(r) which will be given by -

B(r) = B₀ + _$\mu$0 M

B(r) = B₀ + $\chi$(r) B₀

B(r) = B₀ [1 + $\chi$(r)]

B(r) = B₀ [1 + (a r²)]