A sample of paramagnetic salt contains 2.0 × 10²⁴ atomic dipoles each of dipole moment 1.5 × 10⁻²³ J T⁻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

Number of atomic dipoles, n = 2.0 x 10²⁴

Dipole moment of each atomic dipole, M = 1.5 x 10^-23 J T⁻¹

When the magnetic field, Bi = 0.64 T

The sample is cooled to a temperature, T₁ = 4.2°K

Total dipole moment of the atomic dipole,

M_tot = n x M = 2 x 10²⁴ x 1.5 x 10^-23 = 30 J T⁻¹

Magnetic saturation is achieved at 15%.

Hence, effective dipole moment, M₁ = 15/100 x 30 = 4.5 JT^-1

When the magnetic field, B2 = 0.98 T

Temperature, T2 = 2.8°K

Its total dipole moment = M2

According to Curie’s law, we have the ratio of two magnetic dipoles as:

M₂/M₁ = (B₂/B₁ ) x (T₁/T₂)

Therefore, M₂ = (B₂ T₁ M₁)/( B₁T₂)

= (0.98 x 4.2 x 4.5)/(2.8 x 0.64) = 10.336 J T⁻¹

Therefore, 10.336 J T⁻¹ is the total dipole moment of the sample for a magnetic field of 0.98 T and a temperature of 2.8 K.