An iron bar magnet having a coercivity of \(7000 \mathrm{~A} / \mathrm{m}\) is to be demagnetized. If the bar is inserted within a cylindrical wire coil \(0.25 \mathrm{~m}\) long and having 150 turns, what electric current is required to generate the necessary magnetic field?

Coercivity is a property of magnetized materials that measures the resistance of the material to becoming demagnetized. It is defined as the intensity of the magnetic field required to reduce the magnetization of the material to zero after the magnetization has reached saturation.

In essence, coercivity is a measure of the material's 'stickiness' for magnetic alignment—how strongly it holds onto its magnetic field and resists external efforts to demagnetize it. For hard magnetic materials like permanent magnets, coercivity is high which means it takes a substantial magnetic field to demagnetize them. Conversely, soft magnetic materials have low coercivity and are easily demagnetized.

The practical application of understanding coercivity comes into play when trying to demagnetize a material, like an iron bar, as in the exercise provided. Knowing the iron bar's coercivity helps determine the strength of the magnetic field needed to clear any residual magnetic alignment in the bar.

Ampere's law is one of the fundamental equations of electromagnetism that relates the integrated magnetic field around a closed loop to the electric current passing through that loop. The law can be expressed through the formula:

\[\oint B \cdot dl = \mu_0 I_{enc}\]
Here, \(B\) represents the magnetic field, \(dl\) is a small segment of the loop, \(\mu_0\) is the magnetic permeability of free space, and \(I_{enc}\) is the electric current enclosed by the loop.

When dealing with a solenoid, a special form of Ampere's law is used where the magnetic field inside the solenoid is uniform and is given by the formula \(B = \mu_0 n I\), where \(n\) represents the number of turns per unit length of the solenoid. This adaptation is crucial for problems such as finding the current needed to create a magnetic field strong enough to demagnetize a material by reaching its coercivity.

A solenoid is a type of electromagnet that is formed by winding a wire into a tightly packed helix. The concept is a staple in the world of electromagnetism and finds applications in both theoretical and practical scenarios.

The magnetic field within a solenoid is remarkably uniform, and its strength depends on the number of wire turns and the current that flows through the wire. According to Ampere's law, this can be given by \(B = \mu_0 n I\), which we used in our textbook exercise solution. The higher the number of turns per unit length (\(n\)) and the current (\(I\)), the stronger the magnetic field within the solenoid. This property explains how a solenoid can be employed to create a specific magnetic field needed to demagnetize an object by setting its internal magnetic field to the object's coercivity.

Magnetic permeability is a measure of how easily a material can sustain an induced magnetic field inside of it. It's represented by the symbol \(\mu\), and specifically, \(\mu_0\) is the permeability of free space or vacuum.

The value of \(\mu_0 = 4\pi \times 10^{-7} \mathrm{T}\,\mathrm{m} / \mathrm{A}\) is crucial in calculations involving magnetic fields, such as those using Ampere's law. The permeability connects the magnetic field (\(B\)) with the magnetic field intensity (\(H\)) through the relationship \(B = \mu H\).

Materials with high permeability allow magnetic fields to pass through them more easily and are often used in the cores of transformers, electromagnets, and inductors. In the example of the exercise, the iron bar's magnetic permeability would directly influence the effectiveness of the magnetic field produced by the solenoid in the demagnetization process.