Question

A sample of paramagnetic salt contains $2 \times 10^{24}$ atomic dipoles each of dipole moment $1.5 \times 10^{-23}$ JT$^{-1}$. The sample is placed under homogeneous magnetic field of 0.6 T and cooled to a temperature 4.2 K. The degree of magnetic saturation achieved is 20%. Then total dipole moment of the sample for a magnetic field of 0.9 T and a temperature of 2.8 K is

• A. 4.5 JT$^{-1}$
• B. 13.5 JT$^{-1}$
• C. 0.64 JT$^{-1}$
• D. 7 JT$^{-1}$

Hint:

Degree of saturation $\propto B/T$. Total dipole moment = (number of dipoles) $\times$ (average dipole moment per dipole).

Answer 1

The Correct Option is B

The total dipole moment of the sample is given by the product of the number of dipoles and the average dipole moment per dipole. Initially, the degree of magnetic saturation is 20%, which means the average dipole moment per dipole aligned with the field is $0.20 \times 1.5 \times 10^{-23} = 0.3 \times 10^{-23}$ JT$^{-1}$. So the initial total dipole moment is $(2 \times 10^{24})(0.3 \times 10^{-23}) = 6$ JT$^{-1}$. The degree of magnetic saturation is proportional to the ratio $B/T$. Initial ratio $(B/T)_1 = \frac{0.6}{4.2}$. Final ratio $(B/T)_2 = \frac{0.9}{2.8}$. $\frac{(B/T)_2}{(B/T)_1} = \frac{0.9/2.8}{0.6/4.2} = \frac{0.9}{0.6} \times \frac{4.2}{2.8} = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4} = 2.25$ The final degree of saturation will be $20% \times 2.25 = 45%$. So, the final average dipole moment per dipole is $0.45 \times 1.5 \times 10^{-23} = 0.675 \times 10^{-23}$ JT$^{-1}$. The final total dipole moment is $(2 \times 10^{24})(0.675 \times 10^{-23}) = 13.5$ JT$^{-1}$.