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Chapter 20: Problem 6

The magnetization within a bar of some metal alloy is \(3.2 \times 10^{5} \mathrm{~A} / \mathrm{m}\) at an \(H\) field of \(50 \mathrm{~A} / \mathrm{m}\). 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 as being displayed by this material? Why?

Short Answer

Expert verified
Based on the given magnetization (M = 3.2 脳 10鈦� A/m) and magnetic field (H = 50 A/m), we can calculate the magnetic susceptibility (蠂 = 6400), permeability (渭 鈮� 8.025 脳 10鈦宦� Tm/A), and magnetic flux density (B 鈮� 0.401 T) within the metal alloy bar. These calculations suggest that the material most likely exhibits ferromagnetic or ferrimagnetic properties, as it enhances the magnetic field within itself and shows a strong attraction to external magnetic fields. To determine the specific type of magnetism, further characterization is necessary.

Step by step solution

01

Calculate the Magnetic Susceptibility

To calculate the magnetic susceptibility (蠂), we can use the formula 蠂 = M / H. We are given M = 3.2 脳 10鈦� A/m and H = 50 A/m. So, we can plug in the values: 蠂 = (3.2 脳 10鈦� A/m) / (50 A/m) = 6400
02

Calculate the Permeability

Next, we need to calculate the permeability (渭) using the formula 渭 = 渭鈧�(1 + 蠂), where 渭鈧� is the permeability of free space (4蟺 脳 10鈦烩伔 Tm/A). Let's plug in the values: 渭 = (4蟺 脳 10鈦烩伔 Tm/A) 脳 (1 + 6400) 渭 = (4蟺 脳 10鈦烩伔 Tm/A) 脳 6401 鈮� 8.025 脳 10鈦宦� Tm/A
03

Calculate the Magnetic Flux Density

Now, we can calculate the magnetic flux density (B) within the material using the formula B = 渭H. We have 渭 = 8.025 脳 10鈦宦� Tm/A and H = 50 A/m: B = (8.025 脳 10鈦宦� Tm/A) 脳 (50 A/m) 鈮� 0.401 T
04

Determine the Type(s) of Magnetism

Based on our calculations, we can observe the following properties of the metal alloy: 1. The magnetic susceptibility (蠂) is positive and has a large value, which indicates the material is attracted to the magnetic field and enhances the field within itself. 2. The permeability (渭) is greater than the permeability of free space (渭鈧�), suggesting the material is ferromagnetic or ferrimagnetic. 3. The magnetic flux density (B) is also larger than expected for paramagnetic or diamagnetic materials. Considering these observations, we can suggest that this metal alloy displays ferromagnetic or ferrimagnetic properties. Both types of magnetism have a net magnetic moment and are strongly attracted to external magnetic fields. To determine the specific type of magnetism, further characterization such as the temperature dependence of its magnetization or the crystal structure would be needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Susceptibility
Magnetic susceptibility is a measure of how much a material will become magnetized in an applied magnetic field. It is represented by the symbol \( \chi \). A simple way to think about it is how easily the material "listens" or responds to a magnetic field.
  • If \( \chi \) is positive, the material is attracted to the magnetic field.
  • If \( \chi \) is negative, the material is repelled by the magnetic field.
  • Diamagnetic materials have a slight negative \( \chi \), while paramagnetic and ferromagnetic materials have a positive \( \chi \).
For example, in our calculations, a magnetic susceptibility of 6400 indicates that the material significantly enhances the magnetic field applied to it.This is typical of ferromagnetic materials, which can greatly increase a magnetic field within them.
Permeability
Permeability is the measure of a material's ability to support the formation of a magnetic field within itself. It is denoted by the symbol \( \mu \). We use the formula \( \mu = \mu_0 (1 + \chi) \), where \( \mu_0 \) is the permeability of free space, typically valued at \( 4\pi \times 10^{-7} \; \text{Tm/A} \).
  • Materials with high permeability respond strongly to magnetic fields.
  • Ferromagnetic materials have a much higher permeability than air or vacuum.
In our case, the computed permeability of approximately \( 8.025 \times 10^{-3} \; \text{Tm/A} \) indicates a strong ability to support magnetic fields, validating the presence of ferromagnetic or ferrimagnetic properties.
Magnetic Flux Density
Magnetic flux density, represented by \( B \), measures the strength and concentration of a magnetic field. The formula used is \( B = \mu H \), where \( H \) is the magnetic field strength and \( \mu \) is the permeability.
  • Think of \( B \) as the number of magnetic lines per unit area within the material.
  • Higher magnetic flux density means a stronger magnetic presence.
In our example, the magnetic flux density is approximately \( 0.401 \; \text{T} \) (teslas), showing a significant concentration of the magnetic field within the metal alloy. This is another pointer towards strong magnetic properties like those found in ferromagnetism or ferrimagnetism.
Ferromagnetism
Ferromagnetism is a kind of magnetism where certain materials can form permanent magnets. These materials have a high susceptibility and permeability, making them very responsive to magnetic fields.
  • Common ferromagnetic materials include iron, cobalt, and nickel.
  • These materials can remain magnetized even after the external magnetic field is removed.
  • In ferromagnetic materials, magnetic domains line up in the same direction when exposed to a magnetic field, reinforcing each other.
The data from the exercise suggests ferromagnetic properties due to the high susceptibility and permeability found. Ferromagnetic substances are used in everyday items like refrigerator magnets and in electronic devices.
Ferrimagnetism
Ferrimagnetism, while similar to ferromagnetism, involves materials with complex magnetic ordering. Ferrimagnetic materials are typically ceramics like magnetite (Fe鈧僌鈧�).
  • They have regions called domains where the magnetic moments are in opposite alignments but differ in magnitudes.
  • This results in a net magnetic moment, giving them the ability to be strongly magnetic.
  • Ferrimagnetic materials are often used in magnetic storage media and as cores for inductors and transformers.
The potentially high susceptibility in our exploration can also hint at ferrimagnetic properties. To differentiate between ferromagnetic and ferrimagnetic properties in the metal alloy, additional investigations, such as examining the crystal structure and conducting temperature-dependent studies, would be necessary.

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Most popular questions from this chapter

Consult another 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 \(20.4 .\)

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.

Assume there exists some hypothetical metal that exhibits ferromagnetic behavior and that has (1) a simple cubic crystal structure (Figure \(3.24\) ), (2) an atomic radius of \(0.153 \mathrm{~nm}\), and (3) a saturation flux density of \(0.76\) tesla. Determine the number of Bohr magnetons per atom for this material.

A ferromagnetic material has a remanence of \(1.25\) teslas and a coercivity of \(50,000 \mathrm{~A} / \mathrm{m}\). Saturation is achieved at a magnetic field intensity of \(100,000 \mathrm{~A} / \mathrm{m}\), at which the flux density is \(1.50\) teslas. Using these data, sketch the entire hysteresis curve in the range \(H=-100,000 \mathrm{to}+100,000 \mathrm{~A} / \mathrm{m}\). Be sure to scale and label both coordinate axes.

It is possible to express the magnetic susceptibility \(\chi_{m}\) in several different units. For the discussion of this chapter, \(\chi_{m}\) was 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 latter two quantities are related to \(\chi_{m}\) through the relationships $$ \begin{aligned} &\chi_{m}=\chi_{m}(\mathrm{~kg}) \times \text { mass density (in } \mathrm{kg} / \mathrm{m}^{3} \text { ) } \\ &\left.\chi_{m}(\mathrm{a})=\chi_{m}(\mathrm{~kg}) \times \text { atomic weight (in } \mathrm{kg}\right) \end{aligned} $$ When using the cgs-emu system, comparable parameters exist, which may be designated by \(\chi_{m}^{\prime}, \chi_{m}^{\prime}(\mathrm{g})\), and \(\chi_{m}^{\prime}(\mathrm{a})\); the \(\chi_{m}\) and \(\chi_{m}^{\prime}\) are related in accordance with Table 20.1. From Table \(20.2, \chi_{m}\) for silver is \(-2.38 \times 10^{-5}\); convert this value into the other five susceptibilities
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