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Chapter 32: Q35P (page 969)

(a) What is the measured component of the orbital magnetic dipole moment of an electron with $m_{l} = 1 ?$ (b) What is the measured component of the orbital magnetic dipole moment of an electron with $m_{l} = - 2 ?$

Short Answer

Expert verified

(a) The orbital magnetic dipole moment of an electron with $m_{l} = 1$ is $- 9.3 脳 10^{- 24} J / T$

(b) The orbital magnetic dipole moment of an electron with $m_{l} = - 2$ is $1.9 脳 10^{- 23} J / T$

Step by step solution

01

Listing the given quantities

$m_{l} = 1$

$m_{l} = - 2$

02

Understanding the concepts of orbital angular momentum

An electron in an atom has an orbital angular momentum, and for this, we can

associate an orbital magnetic dipole moment. Orbital angular momentum is quantized. By substituting the value of $m_{l}$ , we can find the corresponding component of the orbital magnetic dipole moment.

Formula:

${渭_{o r b}}_{z} = - 渭_{B} m$

03

(a) Calculations of the orbital magnetic dipole moment of an electron with ml=1

${渭_{o r b}}_{z} = - 渭_{B} m_{l}$

For $m_{l} = 1$

role="math" localid="1663153694077" $\begin{array}{rcl} {渭_{o r b}}_{z} & = & - 渭_{B} m_{l} \\ = & - 渭_{B} \\ = & - 9.3 脳 10^{- 24} J / T \end{array}$

The orbital magnetic dipole moment of an electron with $m_{l} = 1$ is $- 9.3 脳 10^{- 24} J / T$

04

(b) Calculations of theorbital magnetic dipole moment of an electron with ml=-2

${渭_{o r b}}_{z} = - 渭_{B} m_{l}$

For $m_{l} = - 2$

$\begin{array}{rcl} {渭_{o r b}}_{z} & = & - 渭_{B} m_{l} \\ = & + 2 渭_{B} \\ = & 2 脳 9.27 脳 10^{- 24} J / T \\ = & 1.9 脳 10^{- 23} J/T \end{array}$

The orbital magnetic dipole moment of an electron with $m_{l} = - 2$ is $1.9 脳 10^{- 23} J / T$

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Most popular questions from this chapter

Consider a solid containing Natoms per unit volume, each atom having a magnetic dipole moment $\overset{鈫�}{渭}$ . Suppose the direction of $\overset{鈫�}{渭}$ can be only parallel or anti-parallel to an externally applied magnetic field (this will be the case if is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy Uis proportional to $\overset{}{e^{\frac{- U}{KT}}}$ , where Tis the temperature and kis Boltzmann鈥檚 constant. Thus, because energy Uis, the fraction of atoms whose dipole moment is parallel to is proportional to $e^{\frac{渭叠}{KT}}$ and the fraction of atoms whose dipole moment is anti-parallel to is proportional to $e^{\frac{- 渭叠}{KT}}$ . (a) Show that the magnitude of the magnetization of this solid is $M = 渭狈 tanh (\frac{渭叠}{kT})$ . Here tanh is the hyperbolic tangent function: $\tanh (x) = (e^{x} - e^{- x}) / (e^{x} + e^{- x})$ (b) Show that the result given in (a) reduces to $M = {狈渭}^{2} B / kT$ for $渭叠鈮� kT$ (c) Show that the result of (a) reduces torole="math" localid="1662964931865" $M = 狈渭$ forrole="math" localid="1662964946451" $渭叠鈮� kT$ .(d) Show that both (b) and (c) agree qualitatively with Figure.

If an electron in an atom has an orbital angular momentum with $m = 0$ , what are the components (a) $L_{o r b, z}$ and (b) $渭_{o r b, z}$ ? If the atom is in an external magnetic field $\overset{鈫�}{B}$ that has magnitude $35 mT$ and is directed along the z axis, what are (c) the energy $U_{o r b}$ associated with $\overset{鈫�}{渭_{o r b}}$ and(d) the energy $U_{s p i n}$ associated with $\overset{鈫�}{渭_{s}}$ ? (e) If, instead, the electron has $m = - 3$ , what are (e) $L_{o r b, z}$ ? (f) $渭_{o r b, z}$ (g) $U_{o r b}$ and (h) $U_{s p i n}$ ?

The figure 32-20 shows a circular region of radius $R = 3.00 cm$ in which adisplacement currentis directedout of the page. The magnitude of the density of this displacement current is $J_{d} = (4.00 A / m^{2}) (1 - r / R)$ , where $r$ is the radial distance $r 鈮� R$ . (a) What is the magnitude of the magnetic field due to displacement current at $2.00 cm$ ? (b)What is the magnitude of the magnetic field due to displacement current at $5.00 cm$ ?

Fig 32-20

Figure 32-41 gives the variation of an electric field that is perpendicular to a circular area of ${2.0m}^{2}$ . During the time period shown, what is the greatest displacement current through the area?

In New Hampshire the average horizontal component of Earth鈥檚 magnetic field in 1912 was $16 渭 T$ , and the average inclination or 鈥渄ip鈥� was $73 掳$ . What was the corresponding magnitude of Earth鈥檚 magnetic field?

See all solutions

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