The magnetic field created by a dipole has a strength of approximately $(μ_{0} / 4 π) (μ / r^{3})$ , where r is the distance from the dipole and $μ_{0}$ is the "permeability of free space," equal to exactly $4 π \times 10^{- 7}$ 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 $μ$ of each dipole is approximately one Bohr magneton $(9 \times 10^{- 24} J / T)$ , 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 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 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 $\partial S / \partial T$ 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 with this material.

Question

The magnetic field created by a dipole has a strength of approximately $(μ_{0} / 4 π) (μ / r^{3})$ , where r is the distance from the dipole and $μ_{0}$ is the "permeability of free space," equal to exactly $4 π \times 10^{- 7}$ 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 $μ$ of each dipole is approximately one Bohr magneton $(9 \times 10^{- 24} J / T)$ , 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 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 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 $\partial S / \partial T$ 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 with this material.

Accepted Answer

a) As a result, the strength of the magnetic field at a dipole's location is determined by the dipoles neighboring is $2.7 \times 10^{- 3} T$ .

b) As a result, the temperature drops by a factor of 370.

c) Thus, without an externally supplied field, the temperature at which this material's entropy grows most quickly as a function of temperature is $1.0 mK$

d) As a result, attempting to achieve a very low temperature using this material would be impractical since heat leakage from the outside cannot be prevented, and a large heat capacity increases the amount of heat necessary to change the temperature.

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