Learning Objectives

➣ Laws of Magnetic Force

➣ Magnetic Field Strength (H)

➣ Magnetic Potential

➣ Flux per Unit Pole

➣ Flux Density (B)

➣ Absolute Parmeability (m) and Relative Permeability (mr)

➣ Intensity of Magnetisation (I)

➣ Susceptibility (K)

➣ Relation Between B, H, I and K

➣ Boundary Conditions

➣ Weber and Ewing’s Molecular Theory

➣ Curie Point. Force on a Current- carrying Conductor Lying in aMagnetic Field

➣ Ampere’s Work Law or Ampere’s Circuital Law

➣ Biot-Savart Law

➣ Savart Law

➣ Force Between two Parallel Conductors

➣ Magnitude of Mutual Force

➣ Definition of Ampere

➣ Magnetic Circuit

➣ Definitions

➣ Composite Series Magnetic Circuit

➣ How to Find Ampere-turns ?

➣ Comparison Between Magnetic and Electric Circuits

➣ Parallel Magnetic Circuits

➣ Series-Parallel Magnetic Circuits

➣ Leakage Flux and Hopkinson’s Leakage Coefficient

➣ Magnetisation Curves

➣ Magnetisation curves by Ballistic Galvanometer

➣ Magnetisation Curves by Fluxmete

6.1. Absolute and Relative Permeabilities of a Medium

The phenomena of magnetism and electromagnetism are dependent upon a certain property of the medium called its permeability. Every medium is supposed to possess two permeabilities :

(i) absolute permeability (μ) and (ii) relative permeability (μr).

For measuring relative permeability, vacuum or free space is chosen as the reference medium. It

is allotted an absolute permeability of μ0 = 4p ´ 10^- henry/metre. Obviously, relative permeability

of vacuum with reference to itself is unity. Hence, for free space,

absolute permeability μ0

= 4p ´ 10^-7 H/m

relative permeability μr = 1.

Now, take any medium other than vacuum. If its relative permeability, as compared to vacuum is

μr, then its absolute permeability is μ = μ0 μr H/m.

6.2. Laws of Magnetic Force

Coulomb was the first to determine experimentally the quantitative expression for the magnetic force between two isolated point poles. It may be noted here that, in view of the fact that magnetic poles always exist in pairs, it is impossible, in practice, to get an isolated pole. The concept of an isolated pole is purely theoretical. However, poles of a thin but long magnet may be assumed to be point poles for all practical purposes (Fig. 6.1). By using a torsion balance, he found that the force between two magnetic poles placed in a medium is

(i) directly proportional to their pole strengths

(ii) inversely proportional to the square of the distance between them and

(iii)

inversely proportional to the absolute permeability of the surrounding medium.

6.1. Magnetic Field Strength (H)

Magnetic field strength at any point within a magnetic field is numerically equally to the force experienced by a N-pole of one weber placed at that point. Hence, unit of H is N/Wb.

Suppose, it is required to find the field intensity at a point A distant r metres from a pole of m

webers. Imagine a similar pole of one weber placed at point A. The force experienced by this pole is

F = m ´ 1 N

Also, if a pole of m Wb is placed in a uniform field of strength H N/Wb, then force experienced by the pole is = mH newtons.

It should be noted that field strength is a vector quantity having both magnitude and direction

It would be helpful to remember that following terms are sometimes interchangeably used with field intensity : Magnetising force, strength of field, magnetic intensity and intensity of magnetic field.

6.1. Magnetic Potential

The magnetic potential at any point within a mag- netic field is measured by the work done in shifting a N-pole of one weber from infinity to that point against the force of the magnetic field. It is given by

6.1. Flux per Unit Pole

A unit N-pole is supposed to radiate out a flux of one weber. Its symbol is F. Therefore, the flux coming out of a N-pole of m weber is given by

F = m Wb

* To commemorate the memory of German physicist Wilhelm Edward Weber (1804-1891).

** A unit magnetic pole is also defined as that magnetic pole which when placed at a distance of one metre from a very long straight conductor carrying a current of one ampere experiences a force of 1/2p newtons (Art. 6.18).

6.1. Flux Density (B)

It is given by the flux passing per unit area through a plane at right angles to the flux. It is usually designated by the capital letter B and is measured in weber/meter². It is a Vector Quantity.

It FWb is the total magnetic flux passing normally through an area of A m², then

B = F/AWb/m² or tesla (T)

Note. Let us find an expression for the flux density at a point distant r metres from a unit N-pole (i.e. a pole of strength 1 Wb.) Imagine a sphere of radius r metres drawn round the unit pole. The flux of 1 Wb radiated out by the unit pole falls normally on a surface of 4pr².m². Hence

6.1. Intensity of Magnetisation (I)

It may be defined as the induced pole strength developed per unit area of the bar. Also, it is the magnetic moment developed per unit volume of the bar.

Let m = pole strength induced in the bar in Wb

A = face or pole area of the bar in m²

Then I = m/A Wb/m²

Hence, it is seen that intensity of magnetisation of a substance may be defined as the flux density produced in it due to its own induced magnetism.

If l is the magnetic length of the bar, then the product (m ´ l) is known as its magnetic moment M.

6.1. Susceptibility (K)

Susceptibility is defined as the ratio of intensity of magnetisation I to the magnetising force H.

\ K = I/H henry/metre.

6.1. Relation Between B, H, I and K

It is obvious from the above discussion in Art. 6.7 that flux density B in a material is given by

B = B0 + m/A = B0 + I \ B = m0 H + I

For ferro-magnetic and para-magnetic substances, K is positive and for diamagnetic substances,

it is negative. For ferro-magnetic substance (like iron, nickel, cobalt and alloys like nickel-iron and cobalt-iron) mr is much greater than unity whereas for para-magnetic substances (like aluminium), µr is slightly greater than unity. For diamagnetic materials (bismuth) µr < 1.

Force on a Current-carrying Conductor Lying in a Magnetic Field

It is found that whenever a current-carrying conductor is placed in magnetic field, it experiences a force which acts in a direction perpendicular both to the direction of the current and the field. In Fig. 6.8 is shown a conductor XY lying at right angles to the uniform horizontal field of flux density B Wb/m² produced by two solenoids A and B. If l is the length of the conductor lying within this field and I ampere the current carried by it, then the magnitude of the force experienced by it is

F = Il B sin 90º = Il B newtons (∵ sin 90º = 1) The direction of this force may be easily found by Fleming’s left-hand rule

Hold out your left hand with forefinger, second finger and thumb at right angles to one another.

If the forefinger represents the direction of the field and the second finger that of the current, then thumb gives the direction of the motion. It is illustrated in Fig. 6.9.

Fig. 6.10 shows another method of finding the direc- tion of force acting on a current carrying conductor. It is known as Flat Left Hand rule. The force acts in the direc- tion of the thumb obviously, the direction of motor of the conductor is the same as that of the force.

It should be noted that no force is exerted on a con- ductor when it lies parallel to the magnetic field. In gen- eral, if the conductor lies at an angle q with the direction of the field, then B can be resolved into two components, B cos q parallel to and B sin q perpendicular to the con- ductor. The former produces no effect whereas the latter is responsible for the motion observed. In that case,

F = BIl sin q newton, which has been expressed as cross product of vector above.*

6.15. Ampere’s Work Law or Ampere’s Circuital Law

The law states that m.m.f.* (magnetomotive force corre- sponding to e.m.f. i.e. electromotive force of electric field) around a closed path is equal to the current enclosed by the path.

Mathematically,

ò H . d s = I amperes where H is the vector

representing magnetic field strength in dot product with vector

d ®s of the enclosing path S around current I ampere and that is

why line integral (“) of dot product

Work law is very comprehensive and is applicable to all magnetic fields whatever the shape of enclosing path e.g. (a) and (b) in Fig. 6.11. Since path c does not enclose the conductor, the m.m.f. around it is zero.

The above work Law is used for obtaining the value of the magnetomotive force around simple idealized circuits like (i) a long straight current-carrying conductor and (ii) a long solenoid.

(i) Magnetomotive Force around a Long Straight Conductor

In Fig. 6.12 is shown a straight conductor which is assumed to extend to infinity in either direction. Let it carry a current of I amperes upwards. The magnetic field consists of circular lines of force having their plane perpendicular to the conductor and their centres at the centre of the conductor.

Suppose that the field strength at point C distant r metres from the centre of the conductor is

H. Then, it means that if a unit N-pole is placed at , it will experience a force of H newtons. The direction of this force would be tangential to the circular line of force passing through C. If this unit N-pole is moved once round the conductor against this force, then work done i.e.

m.m.f. = force ´ distance = I

i.e. I = H ´ 2p r joules = Amperes

6.15. Magnetic Circuit

It may be defined as the route or path which is followed by magnetic flux. The law of magnetic circuit are quite similar to (but not the same as) those of the electric circuit.

Consider a solenoid or a toroidal iron ring having a magnetic path of l metre, area of cross section A m² and a coil of N turns carrying I amperes wound anywhere on it as in Fig. 6.25.

The numerator ‘Nl’ which produces magnetization in the magnetic

Fig. 6.25

circuit is known as magnetomotive force (m.m.f.). Obviously, its unit is ampere-turn (AT)*. It is analogous to e.m.f. in an electric circuit.

The denominator l is called the reluctance of the circuit and is analogous to resistance in

e times, the above equation is called the “Ohm’s Law of Magnetic Circuit” because it resembles a similar expression in electric circuits i.e.

6.15. Definitions Concerning Magnetic Circuit

1. Magnetomotive force (m.m.f.). It drives or tends to drive flux through a magnetic circuit and corresponds to electromotive force (e.m.f.) in an electric circuit.

M.M.F. is equal to the work done in joules in carrying a unit magnetic pole once through the entire magnetic circuit. It is measured in ampere-turns.

In fact, as p.d. between any two points is measured by the work done in carrying a unit charge from one points to another, similarly, m.m.f. between two points is measured by the work done in joules in carrying a unit magnetic pole from one point to another.

2. Ampere-turns (AT). It is the unit of magnetometre force (m.m.f.) and is given by the product of number of turns of a magnetic circuit and the current in amperes in those turns.

3. Reluctance. It is the name given to that property of a material which opposes the creation of magnetic flux in it. It, in fact, measures the opposition offered to the passage of magnetic flux through a material and is analogous to resistance in an electric circuit even in form. Its units is AT/Wb.*

In other words, the reluctance of a magnetic circuit is the number of amp-turns required per weber of magnetic flux in the circuit. Since 1 AT/Wb = 1/henry, the unit of reluctance is “reciprocal henry.”

4. Permeance. It is reciprocal of reluctance and implies the case or readiness with which magnetic flux is developed. It is analogous to conductance in electric circuits. It is measured in terms of Wb/AT or henry.