Magnetism

In a hysteresis loop, when the external magnetizing field (H) is reduced to zero, the remaining magnetic flux density (B) in the material is called Retentivity or Remanence. It represents the ability of a magnetic material to retain its magnetism after the external magnetizing field is removed.

Ferromagnetism is the property of certain materials (like iron, cobalt, and nickel) that causes them to be strongly attracted to an external magnetic field and to retain their magnetic properties even after the external field is removed, thus becoming permanent magnets.
Domain Theory of Ferromagnetism:
The domain theory explains this strong magnetic behavior.
Atomic Magnets: In ferromagnetic materials, individual atoms have a net magnetic moment due to the intrinsic spin of their electrons.
Formation of Domains: Due to a quantum mechanical interaction called exchange coupling, the magnetic moments of adjacent atoms align themselves parallel to each other in large groups. These small regions of spontaneous and uniform magnetization are called magnetic domains. Each domain acts like a tiny, powerful magnet.
Unmagnetized State: In an unmagnetized piece of ferromagnetic material, these domains are oriented randomly. Their magnetic fields point in all different directions, so the net magnetic field of the object is zero.
Magnetization Process: When an external magnetic field is applied, two processes occur:
Domain Growth: Domains that are already aligned (or nearly aligned) with the external field grow in size by moving their boundary walls and taking over adjacent domains that are less favorably aligned.
Domain Rotation: As the external field becomes stronger, the magnetic moments of entire domains that are not yet aligned with the field will rotate to align with it.
Saturated State: When all the domains are aligned with the external field, the material is said to be magnetically saturated. It has reached its maximum possible magnetization.
Permanent Magnetism: When the external field is removed, many of the domains remain locked in their aligned positions due to crystal defects and impurities. This retained magnetism is what makes the material a permanent magnet.

When a bar magnet of magnetic moment and moment of inertia is suspended in a uniform magnetic field , and displaced by a small angle , it experiences a restoring torque .
The restoring torque is given by: For small angular displacements ( ), . This torque is also related to the angular acceleration by the equation . This equation is of the form , which is the condition for angular Simple Harmonic Motion (S.H.M.).
By comparing the two equations, we get the angular frequency : The period of oscillation is related to the angular frequency by .
Substituting the expression for : This is the expression for the period of a bar magnet vibrating in a uniform magnetic field.