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?

Magnetic susceptibility, denoted by the symbol \(\chi\), is a characteristic measure of how much a material can be magnetized by an external magnetic field. The higher the susceptibility, the more a material can be magnetized. It is calculated as the ratio of the material's magnetization \(M\), which is a measure of the magnetic moment per unit volume, to the external magnetic field \(H\).

The calculation of magnetic susceptibility is straightforward. Given the magnetization \(M = 1.2 \times 10^6 \ A/m\) and the applied magnetic field \(H = 200 \ A/m\), the susceptibility \(\chi\) is calculated as \(\chi = M / H\), which turns out to be 6000. This indicates a strong response of the material to the applied magnetic field, suggesting the presence of a ferromagnetic substance because such materials display very high susceptibility.

Ferromagnetic materials are known for retaining magnetic properties even after the external field is removed, which is often linked to their atomic structure that allows magnetic domains to align in the same direction.

Permeability, symbolized by \(\mu\), quantifies a material's ability to conduct magnetic flux. It is fundamentally significant in determining how a magnetic field influences the material and is expressed in terms of Teslas per Ampere-meter (T·m/A). The base reference is the permeability of free space, \(\mu_0\), which is approximately equal to \(4\pi \times 10^{-7} T\cdot m/A\).

To find the permeability of the metal alloy in question, one uses the formula \(\mu = \mu_0(1 + \chi)\). Substituting the values for magnetic susceptibility and the constant \(\mu_0\), we find the permeability to be around \(7.54 \times 10^{-3} T\cdot m/A\). This is substantially higher than the permeability of free space, indicating that this material’s magnetic properties significantly enhance the magnetic flux, which is a hallmark of ferromagnetic substances.

The concept of magnetic flux density, represented by the symbol \(B\), is central to understanding the strength of a magnetic field in a given area. It is defined as the amount of magnetic flux through a unit area in a magnetic field and is measured in Teslas (T). The formula to calculate the magnetic flux density is \(B = \mu H\), which combines both permeability (\(\mu\)) of the material and the external magnetic field (\(H\)).

In our example, with the permeability being \(7.54 \times 10^{-3} T\cdot m/A\) and the external magnetic field \(H = 200 A/m\), the magnetic flux density within the metal alloy is found to be approximately \(1.51 T\). This value indicates a strong internal magnetic field, signifying that the material has a robust capacity to support magnetic flux lines, which is consistent with the properties of ferromagnetic materials that exhibit high flux densities.

Materials exhibit several types of magnetism, each characterized by specific magnetic susceptibilities and permeabilities. The primary categories include:

• Diamagnetism: Found in materials with negative susceptibility, these are weakly repelled by magnetic fields and have a permeability less than \(\mu_0\).
• Paramagnetism: Such materials have a small, positive susceptibility and a permeability slightly greater than \(\mu_0\). They are weakly attracted by magnetic fields.
• Ferromagnetism: Characterized by a large, positive susceptibility and a permeability significantly greater than \(\mu_0\), ferromagnetic materials strongly attract magnetic fields and retain magnetization.

Considering the metal alloy's high magnetic susceptibility and permeability, we can infer that it is likely to be a ferromagnetic material. Ferromagnetism is distinguished by the alignment of atomic magnetic moments in a uniform direction, which results in a strong and permanent magnetization. This property is utilized in a wide range of applications from hard drives to MRI machines. In summary, the metal alloy's behavior in response to external magnetic fields fits the characteristics of ferromagnetism.