RELATION BETWEEN B,H, I AND k

When a magnetic material (say iron) of cross-sectional area A, and relative permeability 𝜇ᵣ is placed in an uniform field of intensity H, two types of lines of induction pass through it: one due to the magnetizing field H and other due to the iron piece itself being magnetized by induction.

Thus total flux density B will be given by

B = 𝜇₀ H + I

where       𝜇₀ = permeability of free space

Now absolute permeability 𝜇ₐ is given by



Dividing either side by 𝜇₀ , we get the relative permeability

MAGNETIC SUSCEPTIBILITY (k)

Magnetic Susceptibility (k) : The Ratio between the intensity of magnetization produced in a substance and the magnetizing force producing it, is called the magnetic susceptibility of the substance, and is denoted by k.

i.e.

INTENSITY OF MAGNETIZATION (I or J)

If an iron piece of area of cross-section A sq meter when placed in an uniform magnetic field of intensity H develops a pole strength of (say) m webers, then the intensity of magnetization I (or J) is defined by

As m is the pole strength induced in the iron piece, the total flux of the piece is m and the flux density is also m/A ; in other words, the intensity of magnetization J, or I, of the iron piece may also be defined as the flux density produced in it due to its own induced magnetism

If l is the length of the iron piece, in meters

where M is the magnetic moment of the piece, and V is its volume in cubic meters.

Thus the intensity of magnetization of a substance can also be defined as the magnetic moment developed per unit volume of the substance.

GAUSS'S THEOREM

Gauss Theorem: Gauss theorem states that "The total number of lines of magnetic induction emanating from a pole to any imaginary closed surface is equal to its pole strength."

Let there be a point pole of strength m webers placed inside a closed surface.

By definition, if B be the flux density over the small elementary area dA sq meters, at a distance r meters, from, the point pole then the flux over the elementary area dA is

If 𝜃 is the angle between the normal to the elementary area dA and the direction of flux, then the flux normal to this elementary area dA is given by

The total flux over the enclosed area is

The factor     is the small solid angle dw subtended by the area dA at the pole.

FIELD OF UNIFORM INTENSITY

Field of Uniform Intensity: A magnetic field in which a unit pole experiences the same force at all points but the direction of force varies from point to point is termed as "Field of Uniform Intensity".

UNIFORM MAGNETIC FIELD

Uniform Magnetic Field: An Uniform Magnetic Field is one in which a pole of say unit strength experiences the same force, is the same.

The lines of force are parallel and straight in an uniform field.

A well known example of an uniform field is magnetic field due to the earth.

MAGNETIC FLUX

Magnetic Flux (𝜙):

In the case of a point pole of strength m webers, the flux density B at distance r is

B=m4πr2   webers/sq meter

∴ Total flux 𝜙 emanating out of strength m webers is

ϕ=flux density×surface area  =B×A

 =m4πr2×4πr2=m webers

ABSOLUTE PERMEABILITY

Definition: The Absolute Permeability (𝜇ₐ) is the ratio of the flux density B (in webers/sq meter) in a substance to the field intensity or magnetizing force H (in newtons/weber of ampere turns/meter) which produces that flux density B.

i.e.,                                                     μa=BH

The Absolute Permeability  𝜇ₐ is also related to the 𝜇₀ (free space) and 𝜇ᵣ (relative permeability) as:

μa=μ0×μr

MAGNETIC FLUX DENSITY (B)

A magnetic pole placed in free space, gives out lines of force in all directions.

The force experienced by a north pole of one weber when placed at P, at a distance r meters from a point pole of m webers, or the field intensity at P, is given by

H=m4π⋅μ0⋅μr⋅r2   N/Wb

The magnitude of field intensity at all points at a distance r from the point pole will be the same, but its direction will be different at different points; but in each case it will be right angles to the surface containing the point.

Now, let us consider a piece of soft iron is placed in a magnetic field H. Let B lines of magnetic induction per square meter of a surface be imagined at right angles to the flux direction.

The magnitude of B is known as the flux density. In  other words, flux density per unit area of a surface at right angles to the flux direction.

Now B will be proportional to H

i.e.,     B∝H      or        B=μ0⋅μr⋅H

Substituting the value of H from above, we have

B=μ0⋅μr⋅m4π⋅μ0⋅μr⋅r2=m4πr2Wb/m2

[Since 𝘮 is in webers and 4𝜋r² is the surface area of a sphere of radius r in meter, therefore B, the flux density is in webers per square meter].