Chapter 4.4, Problem 35P

The magnetic field created by a dipole has a strength of approximately $\left({\mu }_0/4\pi \right)\left(\mu /r^3\right)$, where r is the distance from the dipole and ${\mu }_0$ is the “permeability of free space,” equal to exactly $4\pi \times {10}^{-7}$ in SI units. (In the formula I’m neglecting the variation of field strength with angle, which is at most a factor of 2.) Consider a paramagnetic salt like iron ammonium alum, in which the magnetic moment $\mu$ of each dipole is approximately one Bohr magneton $\left(9\times {10}^{-24}J/T\right)$, with the dipoles separated by a distance of 1 nm. Assume that the dipoles Interact only via ordinary magnetic forces.

(a) Estimate the strength of the magnetic field at the location of a dipole, due to its neighboring dipoles. This is the effective field strength even when there is no externally applied field.

(b) If a magnetic cooling experiment using this material begins with field strength of 1 T, by about what factor will the temperature decrease when the external field is turned off?

(c) Estimate the temperature at which the entropy of this material rises most steeply as a function of temperature, in the absence of an externally applied field.

(d) If the final temperature in a cooling experiment is significantly less than the temperature you found in part (c), the material ends up in a state where $\partial S/\partial T$ is very small and therefore its heat capacity is very small. Explain why it would be impractical to try to reach such a low temperature it this material.