Paramagnetism — Question 7

Question 7

A paramagnetic force thermometer has an inverse-temperature response

A small fixed-volume weak paramagnet has susceptibility $\chi=C/T$ with positive C. At its location the applied field has fixed positive gradient $G=\partial_x(B^2)$. Neglect demagnetization and use $F_x=\chi VG/(2\mu_0)$.

1. Find the force and its temperature derivative at fixed field profile.

2. A measured force drops to 0.80 of its initial value. Find the final-to-initial temperature ratio and the force change for a small fractional temperature increase epsilon.

Question 7 - Solution

A paramagnetic force thermometer has an inverse-temperature response

(a) Method and calculation

$$\boxed{F_x=\frac{CVG}{2\mu_0T}>0}.$$$$\boxed{\frac{dF_x}{dT}=-\frac{CVG}{2\mu_0T^2}=-F_x/T}.$$

(b) Interpretation and further calculation

$$F_f/F_i=T_i/T_f=0.80\Rightarrow\boxed{T_f/T_i=1.25}.$$$$T_f=T_i(1+\varepsilon)\Rightarrow\boxed{\Delta F/F_i\simeq-\varepsilon}\quad(|\varepsilon|\ll1).$$

Physical check

The material is pulled toward stronger field; this inverse-temperature scaling assumes fixed density and an unchanged field gradient.