A sample of a paramagnetic salt contains `2.0 xx 10^(24)` atomic dipoles each of dipole moment `1.5 xx10^(-23) JT ^(-1)` . The sample is placed under a homogeneous magnetic field of 0.64 T and cooled to a temperature of 4.2 K. The degree of magnetic saturation achieved is equal to 15%. What is the total dipole moment of the sample for a magnetic field of 0.98 T and a temperature of 2.8 K ? ( Assume Curie's law)

Explanation

Given, the number of dipoles in the sample
`n = 2.0 xx 10^24`
The magnetic moment of each dipole
`m = 1.5 xx 10^(-23) JT^(-1)`
Initially, the total dipole moment
`M = m xx n = 1.5 xx 10^(-23) xx 2.0 xx 10^24`
` = 30JT^(-1)`
The degree of magnetic saturation achieved is 15%. So, the effective dipole moment is
`M_1 = 15/100 xx 30 = 4.5 JT(-1)`
Using curie.s law `chi = C/T = I/H`
` rArrI = (CH)/(T)`
Magnetic moment is proportional to intensity of magnetization (I), hence for applied field `B (H=B//mu_0)` we can write as follows:
` M prop (CB)/(T )`
` (M_1)/(M_2) = (B_1)/(T_1) xx (T_2)/(B_2)`
GIven `B_1 = 0.84 T, T_1 = 4.2 K, B_2 = 0.98 T " and " T_2 = 2.8 K ` (given)
` therefore M_2 = (M_1T_1B_2)/(B_1T_2)`
` = (4.5 xx .2 xx 0.98)/(0.84 xx 2.8) `
` M_2 = 7.9 JT(-1)`