The total number of lines of force per unit area due to magnetizing field and due to the field induced in the substance is called flux density (B), the unit in which B is measured is Wb/m².
The force experienced by a unit north pole of strength 1 Wb placed at a point in a magnetic field is a measure of the magnetic field intensity due to a pole of strength m at a distance r and is given by the following expression

B⃗ =μ4πmr2r^ Wb/m2

where μ is the absolute permeability of the medium and is expressed as μ=μ0×μr
where μr is relative permeability of the material and μ0 is the permeability of the free space or air and is taken as 4π×10−7Wb/A.m.

Magnetic field strength or Magnetizing field

The magnetic field strength or magnetizing field is given by H⃗ =B⃗ μ A/m and is independent of the medium.
For a coil having n number of turns per unit length and i_o as the (true) current in the winding then
H=nio (ampere-turn/meter)
This value of H is independent of the core material.

Intensity of Magnetization (I or J)

The measure of the magnetization of a magnetized specimen is called intensity of magnetization. It is defined as the magnetic moment per unit volume.

Thus,        I=magneticmomentvolume
As normally the specimen is small its magnetization can be supposed to be uniform. If the specimen is of uniform cross-section a, magnetic length 2l, and pole strength is m, then

M=m×2l and v=a×2l
⸫   I=m×2la×2l=ma Wb/m2

Thus intensity of magnetization is given as pole strength per unit area developed. Its unit is ampere-turn/meter.

Magnetic Susceptibility

The magnetic susceptibility (χ) of a specimen measures the ease with which the specimen can be magnetized and can be defined as the ratio of the intensity of magnetization induced in it and the magnetizing field i.e.
χ=IH

Magnetic Permeability

When a magnetic material is placed in a magnetic field, due to induction it acquires magnetism. The lines of force of the magnetizing field concentrate inside the material and it results in the magnetizing of the material. The measure of the degree to which the lines of force can penetrate or permeate the medium is called absolute permeability of the medium and denoted by μa. The permeability is defined as the ratio of the magnetic induction B in the medium to the magnetizing field H i.e.

μa=μ0μr=B/H

When a magnetic material of cross-sectional area A and relative permeability μr is placed in a uniform field H, two types of lines of induction pass through it, one due to the magnetizing field H and the other due to the material itself being magnetized by induction. Thus the total flux density B will be given by
B=μoH+μoI
As  μa=μoμr=B/H
⸫   BH=μoH+μoIH=μo+μoIH
or   μoμr=μo+μoIH
or   μr=1+χ   or   μaμo=1+χ

Para, Dia, Ferro-Magnetic Substances

Magnetic substances are substances which upon being introduced into an external magnetic field, change so that they themselves become source of an additional magnetic field. Based on their magnetic behaviour substances can be classified into the following three categories.

Paramagnetic Substances

The substances which when placed in a magnetic field acquires a feeble magnetization in the same sense as the applied field are called paramagnetic substances. The examples are platinum, aluminum, manganese, chromium, copper sulphate, iron or nickel salt solutions and crown glass.
Their properties can be summarized as
(i) Such substances in non-uniform magnetic field, experience an attractive force towards the stronger part of the field.
(ii) The permeability m for a paramagnetic substance is slightly greater than one
(iii) The magnetic susceptibility is small positive  value
(iv) For a given temperature χ does not change with variation in H.
(v) The susceptibility varies inversely as the absolute temperature and at higher temperature its value becomes negative.

Diamagnetic substances

The substances, which when placed in a magnetic field acquire feeble magnetization in a direction opposite to that of the applied field are called diamagnetic substances. The examples are bismuth, antimony, water, alcohol and hydrogen. These substances exhibit the following properties.
(i) These substances are repelled by strong magnetic field.
(ii) The permeability μ for diamagnetic substance is less than one but positive.
(iii) Susceptibility for diamagnetic has a small negative value. This value does not vary with field or temperature.
(iv) A diamagnetic substance reduces the flux density B.

Ferromagnetic Substance

Such substances acquire high degree of magnetization in the same sense as the applied magnetic field. The example are: iron, steel, nickel and cobalt. Ferromagnetic substances exhibit the following properties:
(i) They have permeability of the order of hundreds and thousands.
(ii) Susceptibility is also very large and positive.
(iii) For small values of H susceptibility, remains constant and for moderate value of H increases rapidly with H and for large value attains a constant value.
(iv) They are attracted even by weak magnet.
(v) As temperature increases the value of χ decreases.  Above certain temperature ferromagnetic become ordinary paramagnetics and this temperature is called curie temperature (χ∝1T is called curie law). For iron, steel and nickel the curie point is 1000^oC, 770^oC and 360^oC respectively.

Hysteresis Loop

If we take a ferromagnetic material in completely demagnetized state and make it to undergo through a cycle of magnetization in which H is increased from zero to a maximum value H_max, then decreases to zero, then reversed and again taken to –H_max, and finally brought back to zero. The variation of B with respect to H can be represented by a closed Hysteresis loop as shown in figure.To get this graph measure B and H and plot these values. Increase H from zero to H_max and draw the cruve Oa (the maximum value is known as saturation value), this is the normal magnetization curve. Now decrease H from H_max to zero. The induction density B will not fall as rapidly as it increases and will fall back to b rather than zero giving ab as the back trace of Oa. Therefore even when the magnetizing force is made zero or removed, the iron is still magnetic and the flux density Ob is called residual magnetism or retentivity.
Now reverse the magnetizing force H. The value of B becomes zero at point c at which the substance is no longer a magnet. Now H is increased to -H_max and graph cd is obtained. Change –H_max­ to zero  and then to H_max again curve dfa is obtained. This lagging of the flux density B with respect to the magnetizing force H is called hysteresis and the close loop graph is known as hysteresis loop. The value of H required to destroy the residual magnetism is called the coercivity which is represented by Oc.

Energy Loss Due to Hysteresis

To produce a magnetic field a certain amount of energy has to be supplied. This energy is stored in free space where field is established and is returned to circuit when field collapses.
However in case of ferromagnetic substances not all the energy supplied can be returned; part of it is lost in form of heat etc. If the magnetization is carried through a complete cycle, the energy lost is proportional to the area of the hysteresis loop.
When a magnetic material is taken round cycle, there is an energy loss per unit volume of the material given by the area B-H curve.

Properties of Soft and Hard Materials

The shape of the B-H curve depends upon the ferromagnetic substance as shown in figure i.e. it is the characteristic of the substance. From the following figure it can be obtained.
(i) The susceptibility is more for soft materials than for hard material.
(ii) Permeability is more for soft materials than for hard materials.
(iii) Soft materials have greater retentivity as compared to hard materials.
(iv) Hysteresis loss for soft materials is less than that of hard materials.

Earth’s Magnetic Field  (Terrestial Magnetism)

The fact that a freely suspended magnetic compass needle or a bar magnet orients itself roughly along the geographical North-South axis of the earth indicates that there is a magnetic field around the earth. This field is due to circulating electric currents (motion of charged ions of molted substances inside the earth) deep within its interior. We can picture this field as due to a fictitious magnetic dipole deep inside the earth. The earth will not behave like a magnet if it stops rotating.
Components of Earth’s Magnetic Field
Following are three main components of earth’s magnetic field.

Angle of Declination (θ)
A vertical plane passing through N-S line of a freely suspended magnet is called magnetic meridian and the vertical plane passing through the geographical North-South direction is called geographical meridian. The angle of declination is defined at a place as the angle between magnetic meridian and geographical meridian as shown in figure.
Angle of Dip or Inclination (δ)
The total intensity of earth’s magnetic field varies in magnitude as well as in direction from place to place. The angle which the resultant earth’s magnetic field makes with the horizontal line in the magnetic meridian is called magnetic dip or inclination. For measuring this angle we use dip circle and at poles δ=900 and at equator δ=0. (see fig.).

Horizontal Component
The resultant magnetic field due to earth B_e can be resolved into two components (i) horizontal component (H), (ii) vertical component (V). From figure the horizontal component of earths magnetic field is the component of total intensity of earths magnetic field in the horizontal direction in magnetic meridian. i.e.

H=Becosδ   and V=Besinδ