A toroid has a mean radius `R` equal to `20/pi` cm, and a total of 400 turns of wire carrying a current of `2.0 A`. An aluminium ring at temperature `280 K` inside the toroid provides the core. (a) If the magnetization `I` is `4.8 xx 10^(-2) A m^(-1)`, find the susceptibility of aluminium at `280 K`. (b) If the temperature of the aluminium ring is raised to `320 K`, what will be the magnetization?

Explanation

( a) The number of turns per unit length of the toroid is `n=(400)/(2piR)`
The magnetic intensity H in the core is
`H=ni=(400xx2.0A)/(2pixx20/pixx10^(-2)m)=2000A/m`
The susceptibility is
`chi=I//H=(4.8xx10^(-2)A//m)/(2000A//m)=2.4xx10^(-5)`
(b) The susceptibility X of a paramagnetic substance varies with absolute temperature as `chi = c //T`. Thus, `chi_(2)//chi_(1)=T_(1)//T_(2)`
The susceptibility of aluminium at temperature 320 K is, therefore,
`chi=280/320xx2.4xx10^(-5)=2.1xx10^(-5)`
Thus, the magnetization at 320 K is
`I=chiH=2.1xx10^(-5)xx2000A//m=4.2xx10^(-2)A//m`