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
7
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
**
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
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.
5.
Reluctivity.
It is specific reluctance and corresponds to resistivity
which is ‘specific resistance’.
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