} 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 =

An electron is placed in a magnetic field B that is directed along a z axis. The energy difference between parallel and antiparallel alignments of the z component of the electron's spin magnetic moment with B is 5.30 x 10-25 J. What is the magnitude of B ? Number i Units

The Hall effect is to be used to find the density of charge carriers in an unknown material. A Hall voltage 40 uV for 3-A current is observed in a 3-T magnetic field for a rectangular sample with length 2 cm, width 1.5 cm, and height 0.4 cm. Determine the density of the charge carriers.

Electrons enter a magnetic field traveling at v, = vỹ. The constant, uniform magnetic field B = 0.75 2 T acts as a filter curving the electrons to a small gap that only allows specific electron energies. The source of electrons enters the filter as a beam that is 15 micrometers in diameter and have a range of energies from 5-25 meV. The filter redirects the beam, as pictured, to the right v = và but does not speed them up. If the filter is to further narrow the range of energies from 5-25 meV to 10-12 meV, find the dimensions of the gap. In the picture, calculate ymin and ymer, both of which can be assumed to be equal to the radius of the paths taken within the magnetic field. %3D The beam being focused to 15 um at the start can be assumed to be a point source and take circular paths within the magnetic field. Outgoing "Filtered" Electrons Blocked Electrons -不: Y Vma Incoming Electrons