} M/Mmax 1.0 0.75 Curie's law 0.50 0.25 Approximately linear 1.0 2.0 Quantum theory 3.0 4.0 Bext/T (T/K) • 1.30 K • 2.00 K • 3.00 K • 4.21 K A magnetization curve for potassium chromium sulfate, a paramagnetic salt. The ratio of magnetization M of the salt to the maximum possible magnetization Mmax is plotted versus the ratio of the applied magnetic field magnitude Bext to the temperature T. Curie's law fits the data at the left; quantum theory fits all the data. Based on measurements by W. E. Henry. Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment μ. Suppose the direction of μ can be only parallel or antiparallel to an externally applied magnetic field B (this will be the case if μ is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to e-U/kT, where T is the temperature and k is Boltzmann's constant. Thus, because energy U is -μB, the fraction of atoms whose dipole moment is parallel to B is proportional to eμB/T and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e¯µB/kT I NOTE: Give your answers in terms of the variables given. (a) What is the magnitude of the magnetization of this solid? M ( (b) Reduce the the result of expression (a) for µB « kT. M = (c) Reduce the the result of expression (a) for μB » kT. M =
} M/Mmax 1.0 0.75 Curie's law 0.50 0.25 Approximately linear 1.0 2.0 Quantum theory 3.0 4.0 Bext/T (T/K) • 1.30 K • 2.00 K • 3.00 K • 4.21 K A magnetization curve for potassium chromium sulfate, a paramagnetic salt. The ratio of magnetization M of the salt to the maximum possible magnetization Mmax is plotted versus the ratio of the applied magnetic field magnitude Bext to the temperature T. Curie's law fits the data at the left; quantum theory fits all the data. Based on measurements by W. E. Henry. Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment μ. Suppose the direction of μ can be only parallel or antiparallel to an externally applied magnetic field B (this will be the case if μ is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to e-U/kT, where T is the temperature and k is Boltzmann's constant. Thus, because energy U is -μB, the fraction of atoms whose dipole moment is parallel to B is proportional to eμB/T and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e¯µB/kT I NOTE: Give your answers in terms of the variables given. (a) What is the magnitude of the magnetization of this solid? M ( (b) Reduce the the result of expression (a) for µB « kT. M = (c) Reduce the the result of expression (a) for μB » kT. M =
College Physics
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ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
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Transcribed Image Text:}
M/Mmax
1.0
0.75
Curie's
law
0.50
0.25
Approximately linear
1.0
2.0
Quantum theory
3.0
4.0
Bext/T (T/K)
• 1.30 K
• 2.00 K
• 3.00 K
• 4.21 K
A magnetization curve for potassium
chromium sulfate, a paramagnetic salt.
The ratio of magnetization M of the
salt to the maximum possible
magnetization Mmax is plotted versus
the ratio of the applied magnetic field
magnitude Bext to the temperature T.
Curie's law fits the data at the left;
quantum theory fits all the data. Based
on measurements by W. E. Henry.
Consider a solid containing N atoms per unit volume, each atom
having a magnetic dipole moment μ. Suppose the direction of μ can
be only parallel or antiparallel to an externally applied magnetic field
B (this will be the case if μ is due to the spin of a single electron).
According to statistical mechanics, the probability of an atom being
in a state with energy U is proportional to e-U/kT, where T is the
temperature and k is Boltzmann's constant. Thus, because energy U
is -μB, the fraction of atoms whose dipole moment is parallel to B
is proportional to eμB/T and the fraction of atoms whose dipole
moment is antiparallel to B is proportional to e¯µB/kT
I
NOTE: Give your answers in terms of the variables given.
(a) What is the magnitude of the magnetization of this solid?
M
(
(b) Reduce the the result of expression (a) for µB « kT.
M =
(c) Reduce the the result of expression (a) for μB » kT.
M =
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