Consider an ideal air-cored solenoid carrying a steady current I. The magnitude of the magnetic induction inside the solenoid is
where n = \(\cfrac{N}{2\pi r}\) is the number of turns per unit length and p0 is the permeability of free space.
When a core of magnetic material (such as iron) is present, the magnetic field within the solenoid due to the current in the winding magnetizes the material of the core. With the core, the magnetic induction \(\vec B\) inside the solenoid is greater than \(\vec {B_0}\), so that
B = B0 + Bm ………. (2)
where Bm is the contribution of the iron core. Bm is proportional to the magnetization Mz of the material.
Bm = μ0Mz …………… (3)
While discussing magnetic materials, it is customary to call \(\cfrac{B_0}{\mu_0}\) as the magnetizing field or magnetic field intensity, denoted by H, which produces the magnetization.
∴ B0 = μ0H …………. (4)
Substituting for B0 and B in Eq. (1),
B = μ0H + μ0Mz = μ0 (H + Mz) …………….. (5)
For materials in which the magnetization is proportional to the magnetic intensity
Mz ∝ H or Mz = χmH ………. (6)
where the constant of proportionality χm is called the magnetic susceptibility.
∴ B = μ0 (H + χmH) = μ0 (1 + χm)H = μH …………. (7)
where p = μ0 (1 + χm) is called the permeability of the material.
[Notes : (1) Mz ∝ H only for diamagnetic and paramagnetic materials. Among ferromagnetic materials, the linear relation in Eq. (6) holds good only for initial magnetization of magnetically softer materials; for magnetically harder materials, Mz is not a single-valued function of H, and depends on the magnetic intensity that the material has been previously exposed to (a phenomenon known as hysteresis). (2) H is defined for convenience; B is more fundamental.]
