Potassium metal has electron concentration of 1.4 × 1028m-3 and the
corresponding density of states at Fermi level is 6.2 × 1048 Joule-1 m-3. If the
Pauli paramagnetic susceptibility of Potassium is n × 10-L in standard scientific
form, then the value of k (an integer) is __________ (Magnetic moment of
electron is 9. 3 × 10-24 Joule T-1; permeability of free space is 4𝜋 × 10- T m
A-1)
Official Answer Key:
6 to 6
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.29)**
**1. Physics Principles & Pauli Paramagnetism**:
- Given:
- Density of states at Fermi level: $D(E_F) = 6.2 \times 10^{46} \text{ J}^{-1}\text{ m}^{-3}$.
- Magnetic moment of electron: $\mu_B = 9.3 \times 10^{-24} \text{ J T}^{-1}$.
- Permeability of free space: $\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$.
- Formula for Pauli paramagnetic susceptibility:
$\chi_p = \mu_0 \mu_B^2 D(E_F)$
**2. Step-by-Step Numerical Calculation**:
- Calculate $\mu_B^2$:
$\mu_B^2 = (9.3 \times 10^{-24})^2 = 86.49 \times 10^{-48} \text{ J}^2\text{ T}^{-2}$
- Multiply by $D(E_F)$:
$\mu_B^2 D(E_F) = (86.49 \times 10^{-48}) \times (6.2 \times 10^{46}) = 536.238 \times 10^{-2} = 5.362 \text{ J T}^{-2}\text{ m}^{-3}$
- Multiply by $\mu_0 = 4\pi \times 10^{-7} \approx 1.2566 \times 10^{-6}$:
$\chi_p = (1.2566 \times 10^{-6}) \times 5.362 = 6.738 \times 10^{-6}$
- In standard scientific notation: $n \times 10^{-k} = 6.74 \times 10^{-6}$.
- Therefore, the exponent integer is $k = 6$.
**3. Official Verification**:
The verified official IIT Master Answer range is **6 to 6**.
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