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.

Ferromagnetic materials are substances that exhibit a high level of magnetization in response to an external magnetic field and retain their magnetic properties even after the external field is removed. These materials, including iron, nickel, and cobalt, have domains of aligned magnetic moments that contribute to their strong magnetic behavior. When a ferromagnetic material is subjected to a magnetic field, the domains align with the field, increasing the overall magnetization. Upon removing the field, some domains may remain aligned, resulting in a residual magnetic field within the material. This phenomenon is exploited in various applications, such as in the construction of permanent magnets and in data storage technologies where the retention of magnetic state is crucial.

Ferromagnetic materials are essential in understanding the hysteresis curve since their domain alignment and re-orientation under different field strengths give rise to the distinctive looped path characteristic of the curve. These materials enable the curve to show different magnetic properties, such as magnetic remanence and coercivity, which will be discussed further in the following sections.

Magnetic remanence, also known as remanent magnetization, is the magnetization left in a ferromagnetic material after an external magnetic field is removed. It is represented by the symbol \( B_r \) and is measured in tesla (T) in the International System of Units (SI). The presence of magnetic remanence indicates that the material retains a memory of the past magnetic influence, storing magnetic energy in the absence of the field.

In the hysteresis curve, remanence is denoted as the point on the vertical axis where the curve intersects after the external field has reached zero during its decrease from the saturation point. This value of \( B_r \) is crucial when sketching out the curve, as it serves as a starting point for the loop's return path. Understanding \( B_r \) is essential for applications such as magnetic recording, where it is necessary to ensure that data is not easily erased by minor external fields.

Coercivity, noted as \( H_c \) and measured in amperes per meter (A/m), is another critical parameter associated with hysteresis in ferromagnetic materials. It represents the intensity of the applied magnetic field required to reduce the magnetization of the material to zero after the magnetization has been driven to saturation. In other words, it measures the material's resistance to becoming demagnetized.

The value of coercivity can be identified on the hysteresis curve as the point where the curve crosses the horizontal axis, reflecting the field strength necessary to overcome the magnetic remanence. Higher coercivity indicates that a material is better at sustaining its magnetization and can be categorized as a hard magnetic material, which makes it suitable for producing permanent magnets. On the other hand, lower coercivity means that the material is easily magnetized and demagnetized, classifying it as a soft magnetic material, which is ideal for electromagnetic applications where quick changes in magnetization are needed.

Magnetic flux density, commonly denoted by \( B \) and measured in tesla (T), is a measure of the strength and concentration of the magnetic field in a given area. It represents the amount of magnetic flux, the product of the average magnetic field times the perpendicular area it penetrates, through a unit area oriented at ninety degrees to the direction of the magnetic flow.

In the context of the hysteresis curve, the vertical axis typically represents the magnetic flux density, showing how it changes as the external magnetic field varies. A primary feature of the curve is how it rises from the origin to a maximum value called saturation magnetization before descending and creating the loop characteristic of hysteresis. This measurement is essential in the design of electrical and magnetic devices, as it determines how effectively the material can channel magnetic lines of force, impacting the efficiency of transformers, inductors, and motors.

Saturation magnetization refers to the maximum magnetic flux density \( B_{max} \) a ferromagnetic material can achieve. When the material is placed in an external magnetic field and all its magnetic domains are fully aligned, the magnetization plateaus, indicating that the material has reached its full magnetic potential.

This point of saturation is easily identified on the hysteresis curve as it is the peak value along the vertical axis before the loop starts turning. The saturation magnetization not only represents the point at which an increase in the magnetic field will not result in further magnetization but also influences the shape and area of the hysteresis loop. In technical applications, saturation magnetization is an essential aspect when selecting materials for their magnetic performance, ensuring that the components operate within their maximum magnetic capabilities without suffering from saturation losses.