Discussion
The external magnetic field \(B_{z}\) is applied to the paramagnetic material containing N number of spin 1/2 particles each having a magnetic dipole moment \(\mu\). The magnetisation \(\langle M \rangle\) of a system is measured at low magnetic field and at sufficiently high temperate. Which of the following statements is correct ?
Answer: B
The correct statement is \(\langle M \rangle = \frac{N\mu^{2}}{k_{B}T}B_{z}\) with \(\chi \propto 1/T\).
For spin-1/2 moments the two energies are \(\mp\mu B_{z}\), and
\[ \begin{aligned} \langle M \rangle &= N\mu\tanh\frac{\mu B_{z}}{k_{B}T} \\ &\approx \frac{N\mu^{2}B_{z}}{k_{B}T} \end{aligned} \]
in the limit \(\mu B_{z} \ll k_{B}T\) (low field, high temperature), using \(\tanh x \approx x\). The magnetisation is linear in the field, and the susceptibility \(\chi = \partial M/\partial B\) varies as \(1/T\): Curie's law.
\(\chi \propto T\) contradicts the Curie law and goes with the inverted expression in the last option. The option with \(N\mu B_{z}^{2}\) is not linear in the field and has the wrong dimensions.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development