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 \(M\)

Magnetization refers to the extent to which a material becomes magnetized when exposed to a magnetic field. The magnetization of a material is a measure of the density of its induced or permanent magnetic moments. In our exercise, we calculate the magnetization (\(M\)) of chromium when placed within a magnetic field generated by an electric coil.

The formula to calculate magnetization is given by \(M = \chi_m \times H\), where \(\chi_m\) is the magnetic susceptibility of the material (a measure of how much a material will become magnetized in an applied magnetic field), and \(H\) is the magnetic field strength. For chromium, the magnetization turned out to be \(M = 4.08 \, A/m\), when exposed to the magnetic field strength of \(H = 24000 \, A/m\).

This concept is crucial in understanding how materials react to magnetic fields, which is key in many applications ranging from electrical engineering to medical imaging.

Magnetic field strength, commonly represented as \(H\), defines the magnitude of the magnetic field in a coil or magnet. In technical terms, it's the amount of magnetizing force. To find \(H\) for our coil, we use the formula \(H = \frac{NI}{L}\), where \(N\) is the number of turns in the coil, \(I\) is the current running through the coil, and \(L\) is the length of the coil. The coil in our exercise has 400 turns, carries a current of \(15 \, A\), and is \(0.25 \, m\) long. After calculation, we find that the magnetic field strength equals \(24000 \, A/m\).

Understanding the concept of magnetic field strength is essential for students as it applies to fields such as electromagnetism and magnetic circuit design, where it helps calculate the forces exerted by and the performance of magnetic materials.

Susceptibility, denoted as \(\chi_m\), is a dimensionless property that indicates how much a material will become magnetized in the presence of an external magnetic field. It's essentially a ratio indicating the degree of magnetization \(M\) a material obtains in response to an applied magnetic field \(H\).

In our exercise, susceptibility is provided for chromium (\(\chi_m = 1.7 \times 10^{-4}\)) and is used to compute the magnetization of chromium in the coil's magnetic field. This property is key to the field of material science and helps in predicting a material's behavior in a magnetic environment. The susceptibility of chromium suggests that it is only weakly attracted to magnetic fields, which is why the magnetization \(M\) resulted in a relatively small value when compared to \(H\).

Coil current calculation is an essential aspect of understanding electromagnetic coils and their interaction with magnetic fields. The current flowing through a coil (\(I\)) is directly related to the magnetic field strength (\(H\)) it can produce along with the number of turns (\(N\)) and the length of the coil (\(L\)). Using the formula \(H = \frac{NI}{L}\), we can calculate the current required for a targeted magnetic field strength.

In our example, we already know the current is \(15 \, A\), but this formula can be rearranged to solve for \(I\) if we have the desired \(H\) and other variables. Understanding how to manipulate this formula is fundamental for anyone designing electromagnet applications, such as electric motors, transformers, and inductors, where the magnetic field strength must be precisely controlled to achieve desired electrical properties.