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`
`= (400 xx 2.0 A)/(2pi xx 20/(pi) xx 10^(-2) m`
The susceptibility is
`chi = I/H`
`= (4.8 xx 10^(-2) A m^(-1))/(2000 A m^(-1)) = 2.4 xx 10^(-5)`.
(b) The susceptibility `chi` 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 = 280/320 x 2.4 xx 10^(-5) = 2.1 xx 10^(-5)`.
Thus, the magnetization at `320 K` is
`I = chiH`
`2.1 xx 10^(-5) xx 2000 A m^(-1)`.
`4.2 xx 10^(-2) A m^(-1)`.