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Materials Science and Engineering

William D. Callister, David G. Rethwisch

Chapter 20

Magnetic Properties - all with Video Answers

Educators


Chapter Questions

03:38

Problem 1

A coil of wire $0.25 \mathrm{~m}$ long and having 400 turns carries a current of $15 \mathrm{~A}$.
(a) What is the magnitude of the magnetic field strength $H ?$
(b) Compute the flux density $B$ if the coil is in a vacuum.
(c) Compute the flux density inside a bar of chromium positioned within the coil. The susceptibility for chromium is given in Table $20.2$.
(d) Compute the magnitude of the magnetization $M$

Mohammad Mehran
Mohammad Mehran
Numerade Educator
01:56

Problem 2

Demonstrate that the relative permeability and the magnetic susceptibility are related according to Equation $20.7 .$

Ameer Said
Ameer Said
Numerade Educator
01:21

Problem 3

It is possible to express the magnetic susceptibility $\chi_{m}$ in several different units. For the discussion in this chapter, $\chi_{m}$ is used to designate the volume susceptibility in SI units-that is, the quantity that gives the magnetization per unit volume $\left(\mathrm{m}^{3}\right)$ of material when multiplied by $H$. The mass susceptibility $\chi_{m}(\mathrm{~kg})$ yields the magnetic moment (or magnetization) per kilogram of material when multiplied by $H$; similarly, the atomic susceptibility $\chi_{m}$ (a) gives the magnetization per kilogram-mole. The last two quantities
are related to $\chi_{m}$ through the following relationships:
$$
\begin{aligned}
&\chi_{m}=\chi_{m}(\mathrm{~kg}) \times \text { mass density }\left(\text { in } \mathrm{kg} / \mathrm{m}^{3}\right) \\
&\chi_{m}(a)=\chi_{m}(\mathrm{~kg}) \times \text { atomic weight }(\text { in } \mathrm{kg})
\end{aligned}
$$
When using the cgs-emu system, comparable parameters exist that may be designated by $\chi_{m}^{\prime}$, $\chi_{m}^{\prime}(\mathrm{g})$, and $\chi_{m}^{\prime}(a) ;$ the $\chi_{m}$ and $\chi_{m}^{\prime}$ are related in accordance with Table 20.1. From Table 20.2, $\chi_{m}$ for copper is $-0.96 \times 10^{-5}$; convert this value into the other five susceptibilities.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:23

Problem 4

(a) Explain the two sources of magnetic moments for electrons.
(b) Do all electrons have a net magnetic moment? Why or why not?
(c) Do all atoms have a net magnetic moment? Why or why not?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:14

Problem 5

The magnetic flux density within a bar of some material is $0.630$ tesla at an $H$ field of $5 \times 10^{5} \mathrm{~A} / \mathrm{m}$. Compute the following for this material: (a) the magnetic permeability and (b) the magnetic susceptibility. (c) What type(s) of magnetism would you suggest is (are) being displayed by this material? Why?

Ameer Said
Ameer Said
Numerade Educator
02:38

Problem 6

The magnetization within a bar of some metal alloy is $1.2 \times 10^{6} \mathrm{~A} / \mathrm{m}$ at an $H$ field of $200 \mathrm{~A} / \mathrm{m}$. Compute the following: (a) the magnetic susceptibility, (b) the permeability, and (c) the magnetic flux density within this material. (d) What type(s) of magnetism would you suggest is (are) being displayed by this material? Why?

Ameer Said
Ameer Said
Numerade Educator
03:37

Problem 7

Compute (a) the saturation magnetization and (b) the saturation flux density for iron, which has a net magnetic moment per atom of $2.2$ Bohr magnetons and a density of $7.87 \mathrm{~g} / \mathrm{cm}^{3}$

Ameer Said
Ameer Said
Numerade Educator
06:18

Problem 8

Confirm that there are $1.72$ Bohr magnetons associated with each cobalt atom, given that the saturation magnetization is $1.45 \times 10^{6} \mathrm{~A} / \mathrm{m}$, that cobalt has an HCP crystal structure with an atomic radius of $0.1253 \mathrm{~nm}$ and a $c / a$ ratio of $1.623$.

Ameer Said
Ameer Said
Numerade Educator
03:20

Problem 9

Assume there exists some hypothetical metal that exhibits ferromagnetic behavior and that has (1) a simple cubic crystal structure (Figure 3.3), (2) an atomic radius of $0.125 \mathrm{~nm}$, and (3) a saturation flux density of $0.85$ tesla. Determine the number of Bohr magnetons per atom for this material

Ameer Said
Ameer Said
Numerade Educator
02:07

Problem 10

A net magnetic moment is associated with each atom in paramagnetic and ferromagnetic materials. Explain why ferromagnetic materials can be magnetized permanently, whereas paramagnetic ones cannot.

Ameer Said
Ameer Said
Numerade Educator
01:16

Problem 11

Consult a reference in which Hund's rule is outlined, and on its basis explain the net magnetic moments for each of the cations listed in Table $20.4$.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:23

Problem 12

Estimate (a) the saturation magnetization and (b) the saturation flux density of cobalt ferrite $\left[\left(\mathrm{CoFe}_{2} \mathrm{O}_{4}\right)_{8}\right]$, which has a unit cell edge length of $0.838 \mathrm{~nm}$.

Ameer Said
Ameer Said
Numerade Educator
03:24

Problem 13

The chemical formula for copper ferrite may be written as $\left(\mathrm{CuFe}_{2} \mathrm{O}_{4}\right)_{8}$ because there are eight formula units per unit cell. If this material has a saturation magnetization of $1.35 \times 10^{5} \mathrm{~A} / \mathrm{m}$ and a density of $5.40 \mathrm{~g} / \mathrm{cm}^{3}$, estimate the number of Bohr magnetons associated with each $\mathrm{Cu}^{2+}$ ion.

Ameer Said
Ameer Said
Numerade Educator
05:06

Problem 14

The formula for samarium iron garnet $\left(\mathrm{Sm}_{3} \mathrm{Fe}_{5} \mathrm{O}_{12}\right)$ may be written in the form $\mathrm{Sm}_{3}^{c} \mathrm{Fe}_{2}^{a} \mathrm{Fe}_{3}^{d} \mathrm{O}_{12}$, where the superscripts $a, c$, and $d$ represent different sites on which the $\mathrm{Sm}^{3+}$ and $\mathrm{Fe}^{3+}$ ions are located. The spin magnetic moments for the $\mathrm{Sm}^{3+}$ and $\mathrm{Fe}^{3+}$ ions positioned in the $a$ and $c$ sites are oriented parallel to one another and antiparallel to the $\mathrm{Fe}^{3+}$ ions in $d$ sites. Compute the number of Bohr magnetons associated with each $\mathrm{Sm}^{3+}$ ion, given the following information: (1) each unit cell consists of eight formula $\left(\mathrm{Sm}_{3} \mathrm{Fe}_{5} \mathrm{O}_{12}\right)$ units; (2) the unit cell is cubic with an edge length of $1.2529 \mathrm{~nm} ;$ (3) the saturation magnetization for this material is $1.35 \times 10^{5} \mathrm{~A} / \mathrm{m} ;$ and (4) there are 5 Bohr magnetons associated with each $\mathrm{Fe}^{3+}$ ion.

Mohammad Mehran
Mohammad Mehran
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01:51

Problem 15

Briefly explain why the magnitude of the saturation magnetization decreases with increasing temperature for ferromagnetic materials, and why ferromagnetic behavior ceases above the Curie temperature.

Hunza Gilgit
Hunza Gilgit
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02:09

Problem 16

Briefly describe the phenomenon of magnetic hysteresis and why it occurs for ferro- and ferrimagnetic materials.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
05:39

Problem 17

A coil of wire $0.5 \mathrm{~m}$ long and having 20 turns carries a current of $1.0 \mathrm{~A}$.
(a) Compute the flux density if the coil is within a vacuum.
(b) A bar of an iron-silicon alloy, the $B-H$ behavior for which is shown in Figure $20.29$, is positioned within the coil. What is the flux density within this bar?
(c) Suppose that a bar of molybdenum is now situated within the coil. What current must be used to produce the same $B$ field in the Mo as was produced in the iron-silicon alloy (part b) using $1.0 \mathrm{~A}$ ?

Ameer Said
Ameer Said
Numerade Educator
01:49

Problem 18

A ferromagnetic material has a remanence of $1.0$ tesla and a coercivity of $15,000 \mathrm{~A} / \mathrm{m}$. Saturation is achieved at a magnetic field strength of $25,000 \mathrm{~A} / \mathrm{m}$, at which the flux density is $1.25$ teslas. Using these data, sketch the entire hysteresis curve in the range $H=-25,000$ to $+25,000 \mathrm{~A} / \mathrm{m}$. Be sure to scale and label both coordinate axes.

Ameer Said
Ameer Said
Numerade Educator
00:45

Problem 19

The following data are for a plain carbon steel alloy:
\begin{tabular}{cccc}
\hline $\boldsymbol{H}(\boldsymbol{A} / \mathrm{m})$ & $\boldsymbol{B}($ tesla $)$ & $\boldsymbol{H} \mathbf{( A / m )}$ & $\boldsymbol{B}$ (tesla) \\
\hline 0 & 0 & 80 & $0.90$ \\
\hline 15 & $0.007$ & 100 & $1.14$ \\
\hline 30 & $0.033$ & 150 & $1.34$ \\
\hline 50 & $0.10$ & 200 & $1.41$ \\
\hline 60 & $0.30$ & 300 & $1.48$ \\
\hline 70 & $0.63$ & & \\
\hline
\end{tabular}
(a) Construct a graph of $B$ versus $H$.
(b) What are the values of the initial permeability and initial relative permeability?
(c) What is the value of the maximum permeability?
(d) At about what $H$ field does this maximum permeability occur?
(e) To what magnetic susceptibility does this maximum permeability correspond?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:35

Problem 20

An iron bar magnet having a coercivity of $7000 \mathrm{~A} / \mathrm{m}$ is to be demagnetized. If the bar is inserted within a cylindrical wire coil $0.25 \mathrm{~m}$ long and having 150 turns, what electric current is required to generate the necessary magnetic field?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:14

Problem 21

A bar of an iron-silicon alloy having the $B-H$ behavior shown in Figure $20.29$ is inserted within a coil of wire $0.40 \mathrm{~m}$ long and having 50 turns through which passes a current of $0.1 \mathrm{~A}$.
(a) What is the $B$ field within this bar?
(b) At this magnetic field:
(i) What is the permeability?
(ii) What is the relative permeability?
(iii) What is the susceptibility?
(iv) What is the magnetization?

Ameer Said
Ameer Said
Numerade Educator
01:19

Problem 22

Estimate saturation values of $H$ for single-crystal nickel in the [100], [110], and [111] directions.

Ameer Said
Ameer Said
Numerade Educator
02:12

Problem 23

The energy (per unit volume) required to magnetize a ferromagnetic material to saturation $\left(E_{s}\right)$ is defined by the following equation:
$$
E_{s}=\int_{0}^{M_{s}} \mu_{0} H d M
$$
that is, $E_{s}$ is equal to the product of $\mu_{0}$ and the area under an $M$-versus- $H$ curve, to the point of saturation referenced to the ordinate (or $M$ ) axis-for example, in Figure 20.17, the area between the vertical axis and the magnetization curve to $M_{s}$. Estimate $E_{s}$ values (in $\left.\mathrm{J} / \mathrm{m}^{3}\right)$ for single-crystal iron in $[100],[110]$, and $[111]$ directions.

Ameer Said
Ameer Said
Numerade Educator
01:13

Problem 24

Cite the differences between hard and soft magnetic materials in terms of both hysteresis behavior and typical applications.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:03

Problem 25

Assume the silicon-iron (97 Fe, $3 \mathrm{Si}$ ) in Table $20.5$ just reaches the point of saturation when inserted within the coil in Problem 20.1. Compute the saturation magnetization.

Ameer Said
Ameer Said
Numerade Educator
02:54

Problem 26

Figure $20.30$ shows the $B$-versus- $H$ curve for a nickel-iron alloy.
(a) What is the saturation flux density?
(b) What is the saturation magnetization?
(c) What is the remanence?
(d) What is the coercivity?
(e) On the basis of data in Tables $20.5$ and 20.6, would you classify this material as a soft or a hard magnetic material? Why?

Ameer Said
Ameer Said
Numerade Educator
02:20

Problem 27

Briefly explain the manner in which information is stored magnetically.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
00:57

Problem 28

For a superconducting material at a temperature $T$ below the critical temperature $T_{C}$, the critical field $H_{C}(T)$ depends on temperature according to the relationship
$$
H_{C}(T)=H_{C}(0)\left(1-\frac{T^{2}}{T_{C}^{2}}\right)
$$
where $H_{C}(0)$ is the critical field at $0 \mathrm{~K}$.
(a) Using the data in Table $20.7$, calculate the critical magnetic fields for lead at $2.5$ and $5.0 \mathrm{~K}$.
(b) To what temperature must lead be cooled in a magnetic field of $15,000 \mathrm{~A} / \mathrm{m}$ for it to be superconductive?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:16

Problem 29

Using Equation 20.14, determine which of the superconducting elements in Table $20.7$ are superconducting at $2 \mathrm{~K}$ and a magnetic field of $40,000 \mathrm{~A} / \mathrm{m}$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:16

Problem 30

Cite the differences between type I and type II superconductors.

Hunza Gilgit
Hunza Gilgit
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01:11

Problem 31

Briefly describe the Meissner effect.

Hunza Gilgit
Hunza Gilgit
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00:35

Problem 32

Cite the primary limitation of the new superconducting materials that have relatively high critical temperatures.

Ameer Said
Ameer Said
Numerade Educator