Maxwell’s Equations; Magnetism of Matter

Assume that an electron of mass m and charge magnitude e moves in a circular orbit of radius r about a nucleus. A uniform magnetic field $\overrightarrow{B}$ is then established perpendicular to the plane of the orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field $\overrightarrow{B}$ is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.

A sample of the paramagnetic salt to which the magnetization curve of Fig 32-14 applies is to be tested to see whether it obeys Curie’s law. The sample is placed in a uniform 0.50T magnetic field that remains constant throughout the experiment. The magnetization M is then measured at temperatures ranging from 10 to 300K. Will it be found that Curie’s law is valid under these conditions?

A sample of the paramagnetic salt to which the magnetization curve of Figure applies is held at room temperature ( $\mathbf{\text{300 K}}$). (a) At what applied magnetic field will the degree of magnetic saturation of the sample be $\mathbf{\text{50\%}}$ ? (b) At what applied magnetic field will the degree of magnetic saturation of the sample be  $\mathbf{90}\mathit{\%}$? (c) Are these fields attainable in the laboratory?

A magnet in the form of a cylindrical rod has a length of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ and a diameter of $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$. It has a uniform magnetization of $\mathbf{5}\mathbf{.}\mathbf{30}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{A}\mathbf{/}\mathbf{m}$. What is its magnetic dipole moment?

A  $\mathbf{0}\mathbf{.}\mathbf{50}\mathbf{T}$magnetic field is applied to a paramagnetic gas whose atoms have an intrinsic magnetic dipole moment of $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{23}}\mathbf{J}\mathbf{/}\mathbf{T}$. At what temperature will the mean kinetic energy of translation of the atoms equal the energy required to reverse such a dipole end for end in this magnetic field?

An electron with kinetic energy ${\mathbf{K}}_{\mathbf{e}}$ travels in a circular path that is perpendicular to a uniform magnetic field, which is in the positive direction of a z axis. The electron’s motion is subject only to the force due to the field.

(a) Show that the magnetic dipole moment of the electron due to its orbital motion has magnitude $\mathbf{\mu }\mathbf{=}\frac{{\mathbf{K}}_{\mathbf{e}}}{\mathbf{B}}\mathbf{ }$ B and that it is in the direction opposite that of $\overrightarrow{\mathbf{B}}$.(b)What is the magnitude of the magnetic dipole moment of a positive ion with kinetic energy  $\mathbf{ }{\mathbf{K}}_{\mathbf{i}}$ under the same circumstances?  (c) What is the direction of the magnetic dipole moment of a positive ion with kinetic energy $\mathbf{ }{\mathbf{K}}_{\mathbf{i}}$ under the same circumstances? (d) An ionized gas consists of  $\mathbf{5.3}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{21}}$$\mathbf{electron}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$ and the same number density of ions. Take the average electron kinetic energy to be $\mathbf{6}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{20}}\mathbf{ }\mathbf{J}$ and the average ion kinetic energy to be $\mathbf{7}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{21}}\mathbf{J}$. Calculate the magnetization of the gas when it is in a magnetic field of $\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{T}$.

Question: Figuregives the magnetization curve for a paramagnetic material. The vertical axis scale is set by $\mathbf{ }\mathit{a}\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{.}\mathbf{15}$, and the horizontal axis scale is set by $\mathit{b}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{.}\mathbf{2}\mathit{T}\mathbf{/}\mathit{K}$. Let ${\mathit{\mu }}_{\mathbf{s}\mathbf{a}\mathbf{m}}$ be the measured net magnetic moment of a sample of the material and  ${\mathit{\mu }}_{max}$be the maximum possible net magnetic moment of that sample. According to Curie’s law, what would be the ratio ${\mathit{\mu }}_{\mathbf{s}\mathbf{a}\mathbf{m}}\mathbf{/}{\mathit{\mu }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$  were the sample placed in a uniform magnetic field of magnitude, at a temperature of 2.00 k?

Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment $\overrightarrow{\mathbf{\mu }}$. Suppose the direction of  $\overrightarrow{\mathbf{\mu }}$ 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 U is proportional to $\stackrel{}{{\mathbf{e}}^{\frac{\mathbf{-}\mathbf{U}}{\mathbf{KT}}}}$, where T is the temperature and k is Boltzmann’s constant. Thus, because energy U is, the fraction of atoms whose dipole moment is parallel to is proportional to ${\mathbf{e}}^{\frac{\mathbf{\mu B}}{\mathbf{KT}}}$ and the fraction of atoms whose dipole moment is anti-parallel to is proportional to ${\mathbf{e}}^{\frac{\mathbf{-}\mathbf{\mu B}}{\mathbf{KT}}}$. (a) Show that the magnitude of the magnetization of this solid is $\mathbf{M}\mathbf{=}\mathbf{\mu N}\mathbf{ }\mathbf{\text{tanh}}\left(\frac{\mathrm{\mu B}}{\mathrm{kT}}\right)$. Here tanh is the hyperbolic tangent function: $\mathbf{tanh}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{(}{\mathbf{e}}^{\mathbf{x}}\mathbf{-}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{)}\mathbf{/}\mathbf{(}{\mathbf{e}}^{\mathbf{x}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{)}$(b) Show that the result given in (a) reduces to $\mathbf{M}\mathbf{=}{\mathbf{N\mu }}^{\mathbf{2}}\mathbf{B}\mathbf{/}\mathbf{kT}$ for  $\mathbf{\mu B}\mathbf{\ll }\mathbf{kT}$(c) Show that the result of (a) reduces to  $\mathbf{M}\mathbf{=}\mathbf{N\mu }$ for  $\mathbf{\mu B}\mathbf{\gg }\mathbf{kT}$.(d) Show that both (b) and (c) agree qualitatively with Figure.

You place a magnetic compass on a horizontal surface, allow the needle to settle, and then give the compass a gentle wiggle to cause the needle to oscillate about its equilibrium position. The oscillation frequency is $\mathbf{0}\mathbf{.}\mathbf{312}\mathbf{ }\mathbf{Hz}$. Earth’s magnetic field at the location of the compass has a horizontal component of $\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\mu T}$. The needle has a magnetic moment of $\mathbf{0}\mathbf{.}\mathbf{680}\mathbf{ }\mathbf{mJ}\mathbf{/}\mathbf{T}$. What is the needle’s rotational inertia about its (vertical) axis of rotation?

The magnitude of the magnetic dipole moment of Earth is $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}\mathbf{ }\mathbf{J}\mathbf{/}\mathbf{T}$.  (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius?  (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth’s inner core is  $\mathbf{14}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$ .The magnetic dipole moment of an iron atom is  $\mathbf{2}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{23}}\mathbf{ }\mathbf{J}\mathbf{/}\mathbf{T}$.  (Note: Earth’s inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth’s magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)

The magnitude of the dipole moment associated with an atom of iron in an iron bar is $\mathbf{2}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{23}}\mathbf{ }\mathbf{J}\mathbf{/}\mathbf{T}$. Assume that all the atoms in the bar, which is $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$ long and has a cross-sectional area of $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }{\mathbf{cm}}^{\mathbf{2}}$, have their dipole moments aligned. (a) What is the dipole moment of the bar? (b) What torque must be exerted to hold this magnet perpendicular to an external field of magnitude $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{T}$? (The density of iron is $\mathbf{7}\mathbf{.}\mathbf{9}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{m}}^{\mathbf{3}}$.)

The exchange coupling mentioned in Section 32-11 as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, (a) Calculate the magnitude of the magnetic field a distance of $\mathbf{10}\mathbf{ }\mathbf{nm}$away, along the dipole axis, from an atom with magnetic dipole moment  $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{23}}\mathbf{ }\mathbf{J}\mathbf{/}\mathbf{T}$(cobalt), and (b) Calculate the minimum energy required to turn a second identical dipole end for end in this field.  (c) By comparing the latter with the mean translational kinetic energy of  $\mathbf{0}\mathbf{.}\mathbf{040}\mathbf{ }\mathbf{eV}$, what can you conclude?

A magnetic rod with length $\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$, radius $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{mm}$, and (uniform) magnetization  $\mathbf{2.7}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}$A/m can turn about its center like a compass needle. It is placed in a uniform magnetic field $\overrightarrow{\mathbf{B}}$ of magnitude $\mathbf{ }\mathbf{35}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mT}$, such that the directions of its dipole moment and make an angle of $\mathbf{68}\mathbf{.}\mathbf{0}\mathbf{°}$. (a) What is the magnitude of the torque on the rod due to $\overrightarrow{\mathbf{B}}$? (b) What is the change in the orientation energy of the rod if the angle changes to $\mathbf{34}\mathbf{.}\mathbf{0}\mathbf{°}$?

The saturation magnetization ${\mathbf{M}}_{\mathbf{max}}$ of the ferromagnetic metal nickel is $\mathbf{4}\mathbf{.}\mathbf{70}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{A}\mathbf{/}\mathbf{m}$ . Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is  $\mathbf{8}\mathbf{.}\mathbf{90}\mathbf{ }\mathbf{g}\mathbf{/}{\mathbf{cm}}^{\mathbf{3}}$, and its molar mass is $\mathbf{58.71}\mathbf{ }\mathbf{g}\mathbf{/}\mathbf{mol}$ .)

Measurements in mines and boreholes indicate that Earth’s interior temperature increases with depth at the average rate of $\mathbf{30}\mathbf{°}\mathbf{ }\mathbf{C}\mathbf{/}\mathbf{km}$ . Assuming a surface temperature of $\mathbf{10}\mathbf{°}\mathbf{C}$ , at what depth does iron cease to be ferromagnetic? (The Curie temperature of iron varies very little with pressure.)

A Rowland ring is formed of ferromagnetic material. It is circular in cross section, with an inner radius of $\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$  and an outer radius of  $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$, and is wound with $\mathbf{400}$  turns of wire. (a) What current must be set up in the windings to attain a toroidal field of magnitude  ${\mathbf{B}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{mT}$ ?  (b) A secondary coil wound around the toroid has  $\mathbf{50}\mathbf{ }\mathbf{Turns}$ and resistance $\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\Omega }$ . If, for this value of ${\mathbf{B}}_{\mathbf{0}}$ , we have ${\mathbf{B}}_{\mathbf{M}}\mathbf{=}\mathbf{800}\mathbf{ }{\mathbf{B}}_{\mathbf{0}}$  , how much charge moves through the secondary coil when the current in the toroid windings is turned on?

Using the approximations given in Problem 61, find (a) the altitude above Earth’s surface where the magnitude of its magnetic field is 50.0% of the surface value at the same latitude; (b) the maximum magnitude of the magnetic field at the core–mantle boundary, 2900 km below Earth’s surface; and the (c) magnitude and (d) inclination of Earth’s magnetic field at the north geographic pole. (e) Suggest why the values you calculated for (c) and (d) differ from measured values

Earth has a magnetic dipole moment of $\mathbf{\mu }\mathbf{=}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}\mathbf{J}\mathbf{/}\mathbf{T}$. (a) What current would have to be produced in a single turn of wire extending around Earth at its geomagnetic equator if we wished to set up such a dipole? Could such an arrangement be used to cancel out Earth’s magnetism (b) at points in space well above Earth’s surface or (c) on Earth’s surface?

A charge $\mathbf{q}$ is distributed uniformly around a thin ring of radius $\mathbf{r}$ . The ring is rotating about an axis through its center and perpendicular to its plane, at an angular speed $\mathbf{\omega }$ . (a) Show that the magnetic moment due to the rotating charge has magnitude $\mathbf{\mu }\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{q\omega r}}^{\mathbf{2}}$. (b) What is the direction of this magnetic moment if the charge is positive?

A magnetic compass has its needle, of mass $\mathbf{0}\mathbf{.}\mathbf{050}\mathbf{ }\mathbf{kg}$ and length $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$, aligned with the horizontal component of Earth’s magnetic field at a place where that component has the value ${\mathbf{B}}_{\mathbf{h}}\mathbf{=}\mathbf{16}\mathbf{ }\mathbf{\mu T}$. After the compass is given a momentary gentle shake, the needle oscillates with angular frequency $\mathbf{\omega }\mathbf{=}\mathbf{45}\mathbf{ }\mathbf{rad}\mathbf{\backslash }\mathbf{sec}$. Assuming that the needle is a uniform thin rod mounted at its center, find the magnitude of its magnetic dipole moment.

The capacitor in Fig. 32-7 is being charged with a $\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{ }\mathbf{A}$  current. The wire radius is $\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{ }\mathbf{mm}$, and the plate radius is $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ . Assume that the current i in the wire and the displacement current id in the capacitor gap are both uniformly distributed. What is the magnitude of the magnetic field due to i at the following radial distances from the wire’s center: (a) $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mm}$  (inside the wire), (b)  $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mm}$ (outside the wire), and (c)  $\mathbf{2}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{cm}$ (outside the wire)? What is the magnitude of the magnetic field due to id at the following radial distances from the central axis between the plates: (d)  $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mm}$ (inside the gap), (e) $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{mm}$  (inside the gap), and (f)   (outside the gap)? (g) $\mathbf{2}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{cm}$ Explain why the fields at the two smaller radii are so different for the wire and the gap but the fields at the largest radius are not?

A parallel-plate capacitor with circular plates of radius $\mathbf{R}\mathbf{=}\mathbf{16}\mathbf{ }\mathbf{mm}$ and gap width $\mathbf{d}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mm}$  has a uniform electric field between the plates. Starting at time $\mathbf{t}\mathbf{=}\mathbf{0}$, the potential difference between the two plates is $\mathbf{V}\mathbf{=}\mathbf{\left(}\mathbf{100}\mathbf{V}\mathbf{\right)}{\mathbf{e}}^{\mathbf{-}\frac{\mathbf{t}}{\mathbf{\tau }}}$ , where the time constant $\mathbf{\tau }\mathbf{=}\mathbf{12}\mathbf{ }\mathbf{ms}$. At radial distance $\mathbf{r}\mathbf{=}\mathbf{0}\mathbf{.8}\mathbf{ }\mathbf{R}$ from the central axis, what is the magnetic field magnitude (a) as a function of time for  $\mathbf{t}\mathbf{\ge }\mathbf{0}$ and (b) at time $\mathbf{t}\mathbf{=}\mathbf{3}\mathbf{\tau }$?

A magnetic flux of  $\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{mWb}$  is directed outward through the flat bottom face of the closed surface shown in Fig. 32-40. Along the flat top face (which has a radius of  $\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{cm}$ ) there is a  $\mathbf{0}\mathbf{.}\mathbf{40}\mathbf{ }\mathbf{T}$ magnetic field directed perpendicular to the face. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

The magnetic field of Earth can be approximated as the magnetic field of a dipole. The horizontal and vertical components of this field at any distance r from Earth’s center are given by  ${\mathbf{B}}_{\mathbf{H}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{\mu }}{\mathbf{4}{\mathbf{\pi r}}^{\mathbf{3}}}\mathbf{\times }{\mathbf{cos\lambda }}_{\mathbf{m}}\mathbf{ }$, ${\mathbf{B}}_{\mathbf{v}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{\mu }}{\mathbf{2}{\mathbf{\pi r}}^{\mathbf{3}}}\mathbf{\times }{\mathbf{sin\lambda }}_{\mathbf{m}}$ where l_m is the magnetic latitude (this type of latitude is measured from the geomagnetic equator toward the north or south geomagnetic pole). Assume that Earth’s magnetic dipole moment has magnitude $\mathbf{\mu }\mathbf{=}\mathbf{ }\mathbf{ }\mathbf{8.00}\mathbf{ }\mathbf{\times }\mathbf{ }\mathbf{1022}\mathbf{ }\mathbf{A}\mathbf{ }{\mathbf{m}}^{\mathbf{2}}$  . (a) Show that the magnitude of Earth’s field at latitude lm is given by $\mathbf{B}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{\mu }}{\mathbf{4}{\mathbf{\pi r}}^{\mathbf{3}}}\mathbf{\times }\sqrt{\mathbf{1}\mathbf{+}\mathbf{3}{\mathbf{sin}}^{\mathbf{2}}{\mathbf{\lambda }}_{\mathbf{m}}\mathbf{ }\mathbf{ }}$

A parallel-plate capacitor with circular plates of radius   is being charged. At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{\%}$ of its maximum value?

A sample of the paramagnetic salt to which the magnetization curve of Fig. 32-14 applies is immersed in a uniform magnetic field of $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\text{ T}}$. At what temperature will the degree of magnetic saturation of the sample be (a) $\mathbf{50}\mathbf{\%}$ and (b) $\mathbf{90}\mathbf{\%}$
?

A parallel-plate capacitor with circular plates of radius $\mathit{R}$ is being discharged. The displacement current through a central circular area, parallel to the plates and with radius $\raisebox{1ex}{$\mathbf{R}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$, is  $\mathbf{\text{2.0 A}}$. What is the discharging current?

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

What is the measured component of the orbital magnetic dipole moment of an electron with the values (a) $\mathbf{ }{\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{3}$  and (b) ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{4}$?

In Fig. 32-43, a bar magnet lies near a paper cylinder. (a) Sketch the magnetic field lines that pass through the surface of the cylinder. (b) What is the sign of $\overrightarrow{\mathbf{B}}\mathbf{\cdot }\mathit{d}\overrightarrow{\mathbf{A}}$ for every $\mathit{d}\overrightarrow{\mathbf{A}}$ area on the surface? (c) Does this contradict Gauss’ law for magnetism? Explain.

In the lowest energy state of the hydrogen atom, the most probable distance of the single electron from the central proton (the nucleus) is $\mathit{r}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathbf{\text{ m}}$. (a) Compute the magnitude of the proton’s electric field at that distance. The component ${\mathbf{\mu }}_{s,z}$ of the proton’s spin magnetic dipole moment measured on a z axis is $\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{26}}\mathbf{ }\mathbf{ }\raisebox{1ex}{$\mathbf{\text{J}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{\text{T}}$}\right.$. (b) Compute the magnitude of the proton’s magnetic field at the distance $\mathit{r}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathbf{ }\mathbf{ }\mathbf{\text{m}}$ on the z axis. (Hint: Use Eq. 29-27.) (c) What is the ratio of the spin magnetic dipole moment of the electron to that of the proton?

Two plates (as in Fig. 32-7) are being discharged by a constant current. Each plate has a radius of  $\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$. During the discharging, at a point between the plates at radial distance  $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ from the central axis, the magnetic field has a magnitude of  $\mathbf{12}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{nT}$ . (a) What is the magnitude of the magnetic field at radial distance $\mathbf{6}\mathbf{.}\mathbf{00}$  ? (b) What is the current in the wires attached to the plates?

If an electron in an atom has orbital angular momentum with values limited by 3, how many values of (a) ${\mathit{L}}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$ and (b) ${\mathit{\mu }}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$ can the electron have? In terms of h, m, and e, what is the greatest allowed magnitude for (c) ${\mathit{L}}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$ and (d) ${\mathit{\mu }}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$? (e) What is the greatest allowed magnitude for the z component of the electron’s net angular momentum (orbital plus spin)? (f) How many values (signs included) are allowed for the z component of its net angular momentum?

A parallel-plate capacitor with circular plates is being charged. Consider a circular loop centered on the central axis and located between the plates. If the loop radius of $\mathbf{\text{3.00 cm}}$ is greater than the plate radius, what is the displacement current between the plates when the magnetic field along the loop has magnitude $\mathbf{\text{2.00 μT}}$?

Suppose that $\mathbf{4}$  are the limits to the values of an electron in an atom. (a) How many different values of the electrons  ${\mathbf{\mu }}_{\mathbf{orb}\mathbf{,}\mathbf{z}}$ are possible? (b) What is the greatest magnitude of those possible values? Next, if the atom is in a magnetic field of magnitude  $\mathbf{0}\mathbf{.}\mathbf{250}\mathbf{ }\mathbf{T}$, in the positive direction of the z-axis, what are (c) the maximum energy and (d) the minimum energy associated with those possible values of  ${\mathbf{\mu }}_{\mathbf{orb}\mathbf{,}\mathbf{z}}$ ?

What are the measured components of the orbital magnetic dipole moment of an electron with (a) ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{3}$  and (b) ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{4}$?