GATE PH 2022 • Q36 MCQ • 2 Marks

Section 3: Electromagnetic Theory

Electrostatics and Magnetostatics

Subtopic: Biot-Savart Law & Ampere's Circuital Law

Question Statement:

Consider a non-interacting gas of spin 1 particles, each with magnetic moment 𝜇, placed in a weak magnetic field B, such that L ≪ 1. The average magnetic moment of a particle is

Options:

  • (A) $\frac{2\mu}{3} \left(\frac{\mu B}{k_B T}\right)$
  • (B) $\frac{\mu}{2} \left(\frac{\mu B}{k_B T}\right)$
  • (C) $\frac{\mu}{3} \left(\frac{\mu B}{k_B T}\right)$
  • (D) $\frac{3\mu}{4} \left(\frac{\mu B}{k_B T}\right)$

Official Answer Key:

A

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.36)** **1. Statistical Mechanics Principles (Paramagnetism of Spin 1 Particles)**: - Magnetic moment of spin-1 particle: $\vec{\mu} = \mu \vec{S}$. - Under magnetic field $\vec{B} = B\hat{z}$, magnetic quantum number $m_S \in \{-1, 0, +1\}$. - Zeeman energies: $E(m_S) = -m_S \mu B \implies E(+1) = -\mu B, \quad E(0) = 0, \quad E(-1) = +\mu B$ **2. Detailed Step-by-Step Derivation**: - Single-particle partition function: $Z_1 = \sum_{m_S = -1}^1 e^{-\beta E(m_S)} = e^{\beta \mu B} + 1 + e^{-\beta \mu B} = 1 + 2\cosh(\beta \mu B)$ - For weak magnetic field $\beta \mu B = \frac{\mu B}{k_B T} \ll 1$: $\cosh(\beta \mu B) \approx 1 + \frac{(\beta \mu B)^2}{2} \implies Z_1 \approx 3$ $\sinh(\beta \mu B) \approx \beta \mu B$ - Average magnetic moment $\langle \mu_z \rangle$: $\langle \mu_z \rangle = \frac{\mu e^{\beta \mu B} - \mu e^{-\beta \mu B}}{Z_1} = \frac{2\mu \sinh(\beta \mu B)}{1 + 2\cosh(\beta \mu B)}$ - Expanding to first order in $\frac{\mu B}{k_B T}$: $\langle \mu_z \rangle \approx \frac{2\mu (\beta \mu B)}{3} = \frac{2\mu}{3} \left(\frac{\mu B}{k_B T}\right)$ **3. Official Verification**: The verified official IIT Master Answer is **Option (A)** (`2\mu / 3 (\mu B / k_B T)`).

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