GATE 2025 PH – Question 23
A paramagnetic material containing ions with total angular momentum $J=1/2$ is kept at absolute temperature $T$. The ratio of the magnetic field required for 80% of the ions to be in the lowest energy state to that required for having 60% in the lowest energy state at the same temperature is
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Correct answer: (A) $2\ln2/\ln(3/2)$
Explanation
For ions with $J=\tfrac12$ in a magnetic field $B$, there are two levels with energies $\mp\Delta/2$ where $\Delta\propto B$ is the splitting. By the Boltzmann law the ratio of the populations of the lower and the upper level is
$$\frac{p}{1-p}=e^{\Delta/k_BT},$$
where $p$ is the fraction in the lowest state.
**Case 1: $p=0.8$.**
$$\frac{0.8}{0.2}=4=e^{\Delta_1/k_BT}\;\Rightarrow\;\Delta_1=k_BT\ln4 .$$
**Case 2: $p=0.6$.**
$$\frac{0.6}{0.4}=\frac32=e^{\Delta_2/k_BT}\;\Rightarrow\;\Delta_2=k_BT\ln\frac32 .$$
**Ratio of the fields** (the splitting is proportional to $B$):
$$\frac{B_1}{B_2}=\frac{\Delta_1}{\Delta_2}=\frac{\ln4}{\ln(3/2)}=\frac{2\ln2}{\ln(3/2)}\quad(\text{option A}).$$