ELECTROMAGNETISM
THE NATURE OF MAGNETISM
Most electrical equipment depends directly or indirectly upon magnetism. Without magnetismtheworld we perceive today would notexist. There are few electricalusedtoday that do not make use of magnetism.
Natural Magnets
The phenomenon of magnetism was discovered by the Chinese about 2637 B.C. The magnetsused in their primitive compasses were called lodestones or leading stones. It is now known thatlodestones were crude pieces of iron ore known as magnetite. Since magnetite has magneticproperties in its natural state, lode stones are classified as natural magnets. The only other naturalmagnet is the earth itself. All other magnets are human-made and are known as artificial magnets.
Magnetic Fields
Every magnet has two points opposite to each other which most readily attract pieces ofiron. These points are called the poles of the magnet: the North Pole and the South Pole. Justlike electric charges repel each other and opposite charges attractlike magnetic polesrepel each other and unlike poles attract each other. A magnet clearly attracts a bit of iron because of some force that exists around themagnet. This force is called the magnetic field. Although it is invisible to the naked eye, its forcecan be shown to exist by sprinkling small iron filings on a sheet of glass or paper over a bar magnet(Fig. la). If sheet is thetapped gently, the filings will move into a definite pattern whichdescribes the field of force around the magnet. The field seems to be made up of lines of forcethat appearto leave the magnet at the North Pole, travel through the air around the magnet, andcontinue through the magnet tothe South Pole to form a closed loop of force.The stronger the magnet, thegreater the number of lines of force and the larger thearea covered by the field.In order to visualize the magnetic field without iron filings, the field is shown as lines of force inFig. 1b . The direction of the lines outside the magnet shows the path a north pole would followin the field, repelled away from the north pole of the magnet and attracted to itspole.
Figure 1 Magnetic field of force around the magnet
Magnetic Flux
The entire group of magnetic field lines, which flow outward from the north pole of a magnet, iscalled the magnetic flux. The symbol for magnetic flux is the Greek lowercase letter f (phi).The SI unit of magnetic flux is the weber (Wb). One weber equals 1 x 108 magnetic fieldlines.
ELECTROMAGNETISM
In 1819 a Danish scientist named Oersted discovered a relation magnetism and electric current. He found thatan electric current flowing througha conductor produced a magnetic fieldaround that conductor.Fig. 2a filings in a definite pattern of concentric rings around the conductor show the magnetic field of the current in the wire. Every section of the wire has this field offorce around it in a plane perpendicular to the wire (Fig. 2b). The strength of thefieldaround a conductor carrying current dependson the current. A high current will produce many linesof force extending far fromthe wire, while a low current will produceonly a fewclose to the wire(Fig. 9-3).
Figure 2 Circular pattern of magnetic lines around the conductor carrying current
Figure 3 Strength of the magnetic field depends on the amount of current
ELECTROMAGNETIC INDUCTION
In 1831 Michael Faraday discovered the principle of electromagnetic induction. It states thatif a conductor “cuts across” lines of magnetic force, or if lines of force cut across a conductor, anemf, or voltage, is induced across the ends of the conductor. Consider a magnet with its lines offorce extending from the north to the south pole(Fig 3). A conductor C, which can be moved between the poles, is connected to a galvanometer G used to indicate the presence of anemf. When the conductor is not moving, the galvanometer shows zero emf. If the wireconductor is moving outside the magnetic field at position 1, the galvanometer will still show zero. When the conductor is moved to the left to position 2, it cuts across the lines of magneticforce and the galvanometer pointer will deflect to A. This indicates that an emf was inducedin theconductor because lines of force were cut. In position 3, the galvanometer pointer swings backtozero because nolines of force are being cut.Now reverse the direction of the conductor bymoving it right throughthe lines of force back to position 1. During this movement, the pointerwill deflect to B, showing that an emf hasagain been induced in the wire,in the oppositedirection. If the wireis held stationary in the middle of the field of force at position 2, thegalvanometer reads zero. If the conductor is moved up or down parallel to the lines of force so none is cut, no emf is induced.
Fig: 3 When a conductor cuts lines of force, an emf is induced in the conductor
In summary,
1. When lines of force are cut by a conductor or lines of force cut a conductor, an emf, is induced in the conductor.
2. There must be relative motion between the conductor and the lines of force in order toinduce an emf.
3. Changing the direction of cutting will change the direction of the inducedemf.
Faraday’s Laws
1st law: Whenever magnetic flux linking with a coil changes with time an emf is induced in that coil or whenever a moving conductor cuts the magnetic flux, an emf is induced in the conductor.
2nd law: The magnitude of the induced emf is equal to the product of the number of turns of the coil and the rate of change of flux linkage.
Lenz’s law: The direction of the induced emf by electromagnetic induction is in a direction to oppose the main cause producing it.
Explanation:
Consider a coil of N turns and a flux through it changes from an initial value of F1Wb to a final value of F2Wb in time t seconds.
Then, Flux linkages =Number of turns x flux linking with the coil
Now, Initial Flux linkages =NF1and
Final Flux linkages =NF2
Induced e.m.f. = NF1- NF2t1-t2
=NF1- NF2dt
i.e. e = -N dFdt(-) Sign is according to Lenz’s law.
Fleming’s rules:
Fleming’s Right hand rule: This rule helps in deciding the direction of the induced emf.
• Hold the right hand thumb, fore finger and the middle finger set at right angles to each other and the thumb points the direction of the motion of the conductor and the fore fingerpoints the direction of the field and the middle finger points the direction of the induced emf.
Fleming’s Left hand rule: This rule helps in deciding the direction of force acting on a conductor.
• Hold the left hand thumb, fore finger and the middle finger set at right angles to each other and the thumb points the direction of the force acting on the conductor and the direction of the fore finger points the direction of the magnetic field and the middle finger points the direction of the current in the conductor
Induced EMF:
The induced emf in a coil is due to the relative flux cut by it and is of two ways.
Dynamically Induced emf:
consider a stationary magnetic field of flux density B Wb/m2. In this field a conductor with circular cross-section is placed.
Let 'l' be the effective length, in meters, of the conductor in the field.
Let the conductor be moved in the direction i.e. at right angles to the field. In a time dt seconds, the distance moved is 'dx' meters.
Area swept by moving conductor = 1×dx m2
Magnetic flux linked by the conductor = I × dx × B Wb
If the conductor has one turn, the flux linkage F = I× l×dx ×B
Rate of change of flux linkages = dFdt =l×dxdt×B
According to Faraday's law. The emf induced, in the conductor is
e = Bldxdt
i.e.e = BlvVolts, v=linear velocity
The direction of the induced emf is obtained by applying Fleming's right hand rule.
Let the conductor be moved with the same velocity 'v' m/sec in an inclined direction making an angle q to the direction of the field.
Then, the induced emf in the conductor is reduced by sinq.
e = BlvsinqVolts
Statically Induced emf:
In this case, the conductor is held stationary and the magnetic field varied. It may be self-induced or mutually induced.
Consider two coils A and B wound over a magnetic specimen. Coil A is energized using a battery of strength E volts. If switch K is initially closed, then a steady current of I amp will flow through the coil A. It produces a flux of f wb. Let us assume that this entire flux links coils A and B.
When the switch is suddenly opened, the current reduces to zero. Hence, the flux linking both the coils becomes zero. As per Faraday's law, emfs are induced in both the coils A and B. Such emfs are known as statically induced emfs.
Statically induced emf.is also known as "transformer emf." It can be classified into two categories, namely, self induced emf and mutually induced emf.
Self inductance: (L Henry): It is defined as the property of a coil to oppose any change in current.
OR
It is the property of a coil to induce an emf in it when there is a change in current with time.
E= L didt volts
OR
It is the property of a coil to store energy in a magnetic field.
Energy= 12LI2 Joules.
OR
Self inductance is also defined as the change in flux linkage per unit ampere of current change in it.
L= N dfdt from which we write that L=N2S
Equation for Self inductance
N= No.of turns in the magnetizing winding
I=magnetizing current (Amperes)
l=length of the magnetic circuit (m).
a= cross sectional area of the magnetic circuit (m2)
mr is the relative permeability of the specimen.
H= MMFLength=NIl Amp/meter
magnetic flux density, B = momrH=momrNIlTesla
Magnetic flux, j= BA= μOμRNIAlWb
Flux linkage of the circuit = Nj= μOμRN2IAl
Self Inductance, L = NjI =N2Reluctance
Mutual Inductance( M Henry): It is the property of
a coil by which an emf will be induced in it when there
is a change in current in the other neighboring coil wound
on the same core.
e2=M di1dt
inducedemf in the second coil due to a change in the current in the first coil.
From Faraday’s law, e2=N2d∅12dt
Equating the two expressions, M=N2∅12I1 = N1∅21I2
Where, I1 is the current in first coil
I2 is current in second coil
∅12= flux produced in coil 1 linking coil 2
∅21=flux produced in coil 2 linking coil 1
Coupling Coefficient(k):
We know that whenever two coils are placed close to each other, the flux produced in one coil links with the other coil, we say that there is magnetic coupling. But the coupling will not be 100% in general, due to the leakage in flux. To indicate the percentage of flux produced by one coil linking with the other, we use the term coefficient of coupling and it is defined as the ratio of the mutual flux to the total flux produced by a coil and the value is always less than unity.
It is given by k =∅12∅1=∅21∅2
Relationship between Self and Mutual inductance:
We have the mutual inductance, M=N2∅12I1 = N1∅21I2
= N2k∅2I1=N2∅1I2
Also, L1=N1∅1I1
And, L2=N2∅2I2
M2=N2k∅2I1 ×N2∅1I2
=k2 N1∅1I1 N2∅2I2
= k2 L1L2
Thus, M=kÖ(L1.L2)
The self inductance and mutual inductance between the two coils are related as
M=kÖ(L1.L2), where k is the coefficient of coupling.
Leakage flux(Φl): It is defined as the lines of force that are
following an unwanted path.
Expression for the energy stored in a magnetic field.
Let the current through a coil of constant inductance of L henrys growa at a uniform rate from zero to I amperes in time t seconds. Then the average value of the current is (½)I ampere and the emf induced in the coil is (L.I/t) volts. The power absorbed by the magnetic field associated with the coil is the product of the current and the component of the applied voltage is
(1/2)I.(LI)/t watts
and the total energy absorbed by the magnetic field is the product of the average power and time.
Therefore, Energy= (1/2)I.(LI)/t .(t) = ½ LI2 Joules.
Force acting on a current carrying conductor
Whenever a current carrying conductor is subjected to a steady magnetic field there will be a force on the conductor as per Fleming’s left hand rule, and is given by
F=B I l SinqNewtons.
Where I is the current through the coil in amperes, and other terms are the same as mentioned earlier.
SOLVED PROBLEMS
1. A coil of 1500 turns gives rise to a magnetic flux of 2.5 mWb, when carrying a certain current. If the current is reversed in 0.2 sec, what is the average value of the e.m.f. induced in the coil?
Ans: With the reversal of current , there is a reversal of flux too.
change of flux =
= -37.5 volts.
2. A coil of length I am moves at right angles to its length at 60 meter per sec in a uniform magnetic field of density I Wb/m2, Calculate the e.m.f. induced in the conductor, when the direction of motion is (a) Perpendicular to the field (b) Parallel to the field and (c) Inclined at 30° to the direction of the field.
Ans: a) When the conductor moves perpendicular to the direction of the field, e.m.f induced is maximum.
E = B l v Volts
= 1 x 1 x 60 = 60 Volts
b) When the conductor moves parallel to then lines of flux, e.m.f. induced is 0.
c) When the conductor moves at an angle 30°to the direction of the field
e = B l v Sin