Question

A sample of paramagnetic salt contains $2 \times 10^{24}$ atomic dipoles each of dipole moment $1.5 \times 10^{-23} \, \text{JT}^{-1}$. 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:

Paramagnetic material magnetization is proportional to B/T.

Answer 1

The Correct Option is B

The total dipole moment of the sample is proportional to the magnetic field strength (B) and inversely proportional to the temperature (T). Since the degree of magnetic saturation is 20% at 0.6 T and 4.2 K, the effective number of dipoles contributing to the magnetization is $2 \times 10^{24} \times 0.2 = 4 \times 10^{23}$. The total dipole moment at these conditions is $4 \times 10^{23} \times 1.5 \times 10^{-23} = 6 \, \text{JT}^{-1}$. Now, we want to find the total dipole moment at 0.9 T and 2.8 K. Let $M_1$ be the total dipole moment at 0.6T and 4.2 K, and $M_2$ be the total dipole moment at 0.9 T and 2.8 K. Since the number of dipoles is constant, we have: $$ \frac{M_2}{M_1} = \frac{B_2/T_2}{B_1/T_1} = \frac{B_2 T_1}{B_1 T_2} $$ Plugging in the values, we get: $$ \frac{M_2}{6} = \frac{0.9 \times 4.2}{0.6 \times 2.8} = \frac{3.78}{1.68} = 2.25 $$ Therefore, $M_2 = 6 \times 2.25 = 13.5 \, \text{JT}^{-1}$.