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?

Understanding magnetic materials often requires examining how they respond to an external magnetic field. The hysteresis loop is a graph that represents the relationship between the magnetic flux density, denoted as 'B', and the magnetic field intensity, symbolized as 'H'. If we were to create a hysteresis loop for the provided data, we would plot 'B' versus 'H' as instructed in the exercise.

When you first apply an increasing magnetic field to a magnetic material, 'B' increases as well until it reaches a saturation point where the material cannot be magnetized any further. Once the external field is decreased, the path that 'B' follows doesn't retrace its initial steps but rather loops around, leading to the phenomenon known as hysteresis. This loop illustrates the material's retentivity, which is the ability to retain magnetization, and coercivity, which is the resistance to becoming demagnetized.

The hysteresis loop is critical in understanding magnetic materials because it reveals properties such as energy loss during magnetic cycling, which is important in various applications like electrical transformers and memory devices.

Magnetic flux density 'B', also known as magnetic induction, is essentially the strength of the magnetic field in a material, representing the amount of magnetic flux in a unit area perpendicular to the direction of magnetic flow. It is measured in teslas (T) and, as seen in the exercise, is graphed against 'H' to help understand the material's magnetic properties.

Magnetic flux density is a fundamental concept in electromagnetism that helps us not only visualize but also calculate the forces that occur in magnetic fields. For a given area, 'B' can tell us how much magnetic flux we're dealing with, which is crucial in applications such as electric motors, generators, and sensors where magnetic fields play a crucial role.

Relative permeability ('µr') is a dimensionless measure that expresses how easily a material can become magnetized relative to the vacuum of free space. It's a ratio defined by the permeability of the material ('µ') divided by the permeability of free space ('µ0'). The relative permeability helps us to categorize materials as diamagnetic, paramagnetic, or ferromagnetic based on their response to magnetic fields.

For instance, materials with a 'µr' less than 1 are considered diamagnetic and tend to repel magnetic fields, whereas materials with a 'µr' greater than 1 are either paramagnetic or ferromagnetic, meaning they are attracted to magnetic fields to varying degrees. The concept of relative permeability is significant in the design of electromagnetic devices, as it affects how well a material can channel magnetic lines of force.

Magnetic field intensity 'H' portrays the strength of a magnetic field caused by an electrical current or a magnetizing force. It's measured in amperes per meter (A/m) and is essentially a way of quantifying the 'push' or driving force behind the magnetic field. When we talk about 'H', we refer to the applied magnetic field that causes a material to become magnetized.

It is essential to differentiate between 'B' and 'H'; where 'B' describes the total magnetic field including the material's response, 'H' focuses on the external influence or the initially applied field. Understanding 'H' assists in predicting how materials will behave in a magnetic circuit, which is indispensable in creating and evaluating magnetic systems like solenoids and electromagnets.

Magnetic susceptibility ('χm') is a dimensionless parameter indicating how susceptible a material is to becoming magnetized when placed within a magnetic field. It is intimately related to relative permeability, as exemplified in the exercise where the susceptibility is found using the formula \(χm = µr - 1\).

Magnetic susceptibility allows us to understand the degree to which a material will magnetize in response to an applied field, essentially indicating how 'willing' the material is to adopt the magnetic field lines. Its value can be positive for materials that are attracted to magnetic fields and negative for those that are repelled. This property is particularly relevant when classifying materials and also in magnetic resonance imaging (MRI) technology where it helps in distinguishing between different types of tissues within the human body.