PROBLEM 35P

The magnetic field created by a dipole has a strength of approximately (μ0/4π)(μ/r3), where r is the distance from the dipole and μ0 is the “permeability of free space,” equal to exactly 4π × 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 μ of each dipole is approximately one Bohr magneton (9 × 10–24 J/T), 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 neighbouring 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 an external 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 ∂S/∂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 with this material.