Question

A sample of paramagnetic salt contains $2.0\times {10}^{24}$ atomic dipoles each of dipole moment $1.5\times {10}^{-23}J/T$.. The sample is placed under a homogeneous magnetic field of $0.64T$ and cooled to a temperature of $4.2K$. The degree of magnetic saturation achieved is equal to $15\%$. What is the total dipole moment of the sample for a magnetic field of $0.98T$ and a temperature of $2.8K$? (Assume Curie’s law)

Answer 1

Step 1: Find the total dipole moment of the sample.

Here no. of dipoles $n=2\times {10}^{24}$
Magnetic moment of each dipole $M=1.5\times {10}^{-23}J/T$
Total dipole moment of sample $=n\times M=2\times {10}^{24}\times 1.5\times {10}^{-23}J/T$

Step 2: Find the total dipole moment when saturation achieved $15\%$
As saturation achieved is $15\%$ therefore, effective dipole moment

$M_1=\frac{15}{100}\times 30=4.5J/T;B_1=0.64T,T_1=4.2K$

$M_2=?;B_2=0.98T,T_2=2.8K$

$=\frac{15}{100}\times 30=4.5J/T;B_1=0.64T,T_1=4.2K$

$M_2=?;B_2=0.98T,T_2=2.8K$

The ratio of magnetic dipole from Curie temperature, $\frac{M_2}{M_1}=\frac{B_2}{B_1}\times \frac{T_1}{T_2}$

$M_2=M_1\frac{B_2}{B_1}\times \frac{T_1}{T_2}$

$M_2=4.5\frac{0.98}{0.64}\times \frac{4.2}{2.8}$

$M_2=10.34J/T$

Final Answer: $10.34J/T$