Question

Figure 32-22a shows a pair of opposite spins orientations for an electron in an external magnetic field${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$. Figure 32-22b gives three choices for the graph of the energies associated with those orientations as a function of the magnitude${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$. Choices b and c consist of intersecting lines and choice of parallel lines. Which is the correct choice?

Short answer

The correct choice for the graph of the energies associated with orientations as a function of${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$ is b.

Step-by-step solution

The given data

Figure 32-22a with two opposite spin orientations of an electron in a magnetic field is given.

Understanding the concept of the potential energy due to spin orientations

The potential energy of an electric dipole is given as the amount of work done on the dipole to rotate in from an initial zero potential energy position to any desired potential energy position. The electron spin orientation determines the direction of the magnetic dipole that is used by the right-hand curl rule to determine the magnitude of the energy respective to the uniform magnetic field. Thus, the lowest value of energy is obtained when the dipole is aligned with the magnetic field. Using the formula for the potential for electrons,the correct choice for the graph of the energies associated with orientations as a function${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$can be found.

Formula:

$\mathit{U}\mathbf{=}\mathbf{-}\stackrel{\mathbf{\rightharpoonup }}{{\mathbf{\mu }}_{\mathbf{s}}}\mathbf{.}{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$………………………….. (i)

where, U is the potential energy of a system, B is the external magnetic field of the system, is the spin magnetic moment of a charged particle.

Determining the correct choice for the graph of the energies associated with orientations as a function of B⇀ext

From equation (i), the relation of the potential energy to the dipole moment can be given as:

$U\propto -\stackrel{\rightharpoonup }{{\mu }_s},$

Again, the direction of the dipole moment is given as the direction of the electron spin.

Thus, for two opposite spins, the value of dipole moments varies equally opposite.

Now, for the spin direction parallel to the magnetic field, the potential energy will be given using equation (i) as:

$U=-{\mu }_s{B}_{ext}$

Again, for the spin direction anti-parallel to the magnetic field, the potential energy will be given using equation (i) as:

$U=-{\mu }_s{B}_{ext}$

Thus, $-\stackrel{\rightharpoonup }{{\mu }_s}$is positive for one orientation and negative for one orientation.

This is depicted in the slope of graph b.

Therefore, the correct choice for the graph of the energies associated with orientations as a function of ${\stackrel{\rightharpoonup }{B}}_{ext}$is b.