A coil of wire \(0.20 \mathrm{~m}\) long and having 200 turns carries a current of \(10 \mathrm{~A}\). (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 The susceptibility for titanium is found in Table \(20.2\). (d) Compute the magnitude of the magnetization \(M\).

Magnetic flux density, commonly represented by the symbol \(B\), is a measure of the strength of the magnetic field in a particular area and is defined as the force exerted on a moving charge in a magnetic field. Think of it like the density of the magnetic 'lines of force' in a given space. The formula \(B = \mu_0 * H\) applies in a vacuum, where \(\mu_0\) is the permeability of free space and \(H\) is the magnetic field strength. This concept explains that in a vacuum, the magnetic flux density is proportional to the magnetic field strength. This means magnetic flux density increases with the increase in magnetic field strength.

The exercise provided an example where the coil generates a magnetic field, and the flux density in a vacuum was calculated using the formula given above. The assumption of a vacuum is important since it implies that there are no other materials influencing the magnetic field. This serves as a foundation for understanding magnetic fields in more complex environments, such as within different materials.

Magnetization, denoted by \(M\), refers to the degree to which a material becomes magnetized when it is placed within a magnetic field. It is essentially the magnetic moment per unit volume of the material and indicates how strongly the material responds to the magnetic field. Intuitively, it's a measure of how much 'magnetic stuff' is in a given volume of material. The formula for calculating magnetization is \(M = H * \chi\), where \(H\) represents the magnetic field strength, and \(\chi\) is the material's susceptibility.

In the exercise, magnetization is calculated for a bar of titanium placed within the coil's magnetic field. The susceptibility in the formula represents the titanium’s ability to become magnetized. When a material with a certain susceptibility is placed within a magnetic field, its magnetization can be computed to see just how magnetic it becomes. This is crucial in applications like magnetic storage devices, transformers, and electric motors, where material magnetization is a key factor.

Susceptibility, symbolized by \(\chi\), is a dimensionless constant that describes how much a material can be magnetized in response to an external magnetic field. It's a coefficient that reveals how easily a material becomes magnetized. When susceptibility is positive, the material is termed 'paramagnetic', which means it enhances the external magnetic field; when negative, it's 'diamagnetic', tending to weaken the external field. If the susceptibility is large (either positively or negatively), the material's magnetic properties are more pronounced.

In the step by step solution of the exercise, the susceptibility of titanium is used to modify the magnetic flux density formula. As susceptibility changes from one material to another, understanding this property helps in selecting appropriate materials for various technology applications. For example, materials with high susceptibility are used in inductors and transformers to help generate stronger and more efficient magnetic fields.

Understanding Ampere’s Law

Ampere's Law is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is mathematically represented by \(\oint B \cdot dl = \mu_0 I_{enc}\), where \(\oint B \cdot dl\) is the magnetic field around a loop, \(I_{enc}\) is the current enclosed by the loop, and \(\mu_0\) is the permeability of free space.

Ampere’s Law helps to predict the magnetic field created by electric currents in structures such as coils and solenoids. It is often used in conjunction with the Biot-Savart Law for calculating magnetic fields. In practical scenarios, such as in the design of electromagnets or inductive components, Ampere's Law is instrumental in understanding how current creates a magnetic field, and how to manipulate that field by changing the current or the shape of the wire carrying the current.

In essence, Ampere's Law explains how currents generate magnetic fields, which is essential in creating and controlling magnetic fields in a variety of electrical and mechanical systems.