4 MOVING CHARGES AND MAGNETISM. This force provides centripetal force for the particle and it moves in a circular path.

Transcription

1 4 MOVING CHARGES AND MAGNETISM Magnetic Lorentz force When a charged particle moves with a velocity v in a magnetic field of strength B, a magnetic Lorentz force acts on it. F = qvb Sinθ = q( v B) It acts perpendicular to velocity (or displacement). So the work done by magnetic Lorentz force on a moving charge is zero. Motion of a charged particle in a direction perpendicular to a magnetic field When a charged particle moves with a velocity v in a magnetic field of strength B, a magnetic Lorentz force acts on it. F = qvb Sinθ F = qvb Here θ = 90 o This force provides centripetal force for the particle and it moves in a circular path. qvb = or r = = r mv ie, Radius is proportional to momentum. Time period of particle T = = π = π = π Angular frequency of particle, ω = 2π = 2π ω = 2π π ω = The SI unit of magnetic induction (magnetic field intensity or magnetic flux density) is tesla(t) or weber /m 2. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 1

2 Motion of a charged particle at an angle with a magnetic field Consider a charged particle moving at an angle with the magnetic field. The velocity of the particle can be resolved into two components -one component parallel to B and one component perpendicular to B. As there is a component of velocity perpendicular to magnetic field, the particle moves in a circular path. No force acts in the horizontal direction as the angle between B and horizontal component of velocity is zero. Therefore the path becomes helical. The distance moved by the particle along the field in one rotation is called pitch. When a charged particle enters perpendicular to a uniform magnetic field, the speed and kinetic energy of the particle in the circular path remains the same. Velocity changes and momentum also changes (due to change of direction). When two different charged particles (electron and proton) having same momentum enter perpendicular to a uniform magnetic field, their paths are equally curved (since radius is proportional to momentum). An electron is not deflected through a certain region of space. Can we say that there is no magnetic field in that region? No. Electron experiences no force when it moves parallel or anti-parallel to the magnetic field. Electric field can exert force on a charge at rest or in motion. Magnetic field can exert force only if the charge is moving. The magnetic force on a charge moving perpendicular to a magnetic field can be found by using Fleming s left hand rule Stretch the fore finger, middle finger and thumb of left hand in mutually perpendicular directions. If fore finger represents magnetic field and middle finger represents the direction of motion of positive charge (current), then the thumb represents the force. PHYSICS It can be found by using right hand palm rule also. Open up the right hand palm. If the thumb represents the direction of motion of positive charge (current), the fingers point in the direction of current then palm faces towards force. GK's Unique Learning Centre, Ulloor, Tvpm. Mob: Page 2

3 Prove that the work done by magnetic force on a charged particle entering the field is zero. The magnetic force acting on a charged particle entering a magnetic field is = q ( ) is perpendicular to both. As is perpendicular to. = 0 But. = 2 m = 0 or. = 0 m. = = 0 = 0 As K is constant, the magnitude of velocity is also constant. F = ma = m m. = 0. = 0 = 0 K = 0 or K (Kinetic energy) is a constant. By work energy theorem, Work done = Change of kinetic energy Here Change of kinetic energy = 0 Work done is zero. Motion of a charged particle in a combined electric and magnetic field When a charge moves in a combined field, Force exerted by electric field = Eq Force exerted by magnetic field = q ( v B) Total force = Eq + q( v B) If the fields are perpendicular and also perpendicular to the velocity of particle as in figure, E = Eȷ, B = Bk, v = vı F e = qe = qeȷ (along y direction) F B = q( v B) = q(vı Bk) = qvbı k = qvb( ȷ ) = qvbȷ (along y direction) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 3

4 If the magnitude of the force are equal, qvb = Eq or vb = E or v = If a beam containing charged particles are sent to a crossed region of electric and magnetic fields, the particle with velocity = passes through the fields. So the fields act as velocity selector. Velocity selector is otherwise called velocity filter. Aurora borealis and aurora australis The electrons and protons of solar wind are trapped near the earth s poles. These particles spiral around the magnetic lines. They bounds back and forth and emit light (green pink colour). It produces a beautiful glow of colours in the sky. This is called aurora. The aurora in the north arctic region is called northern light or aurora borealis. The aurora in the south is called aurora australis. Mass spectrometer is the device used to separate charged particles. Working of cyclotron Cyclotron is a particle accelerator. It was devised by Lawrence and Livingstone. It is used to accelerate positively charged particles like protons, deuterons, alpha particles etc. It consists of two semi-circular metal boxes called dees D 1 and D 2. These are connected to a high frequency alternating source. At the centre, there is a positive ion source. A magnetic field is applied perpendicular to the flat surface of dees. The whole set up is kept inside an evacuated chamber. The ion source emits ions. Suppose D 1 is negative. The ion is attracted towards it. As the ion enters perpendicular to magnetic field, it moves in a circular path. Inside the dees the particle (ion) is not affected by electric field. When it completes a half rotation, the polarity of dees are reversed. When D 2 becomes negative, the ion is attracted towards D 2. When the ion is in the gap between dees, it is accelerated due to the effect of electric field. Its kinetic energy increases. The radius of circular path increases and finally it is taken out with the help of a deflector. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 4

5 The frequency of external source is made equal to the frequency of revolution of the particle. Then only the polarity of dees get reversed when the particle completes a half rotation. This frequency is called cyclotron frequency or magnetic resonance frequency. Time period of particle = But r = T = T = Frequency of particle = π π π = π This will be equal to the frequency of alternating source. Cyclotron frequency = π It is clear that time period and frequency are independent of velocity or radius of orbit. Velocity of the particle = Kinetic energy of the particle = Limitations mv = m It cannot be used to accelerate uncharged particles. = Lighter particles like electrons cannot be accelerated using cyclotron. When the velocity increases, mass of these particles increases. Therefore frequency decreases. Frequency of alternating source cannot be changed in accordance with this decrease of frequency. Note: In a cyclotron, Electric field accelerates the positively charged particles Magnetic field keeps the particle in circular path Magnetic force on a current carrying conductor kept in a magnetic field Consider a conductor of length and area of cross section A containing n electrons per unit volume. Volume of the conductor Number of electrons in the conductor If the charge of one electron is q, the total charge = A = An = Anq When this conductor is connected to a battery and kept in a magnetic field, The force exerted by the magnetic field on each electron of the conductor = q( v B ) Force on all the electrons = Anq( v B ) But ı = vqna or ȷ = vqn Force = (A J B ) ` But = A J = I Force = I B (Transferred the vector sign from J to ) Force exerted by magnetic field, F = I B Unique Learning Centre, Ulloor, Tvpm. Mob: Page 5

6 Here is a vector of magnitude equal to the length of conductor and whose direction is the same as that of. Force on a current carrying conductor kept in a magnetic field if current is perpendicular to magnetic field is given by Fleming s left hand rule. Biot Savart s law and its mathematical form Magnetic field at a point (db ) due to a small element of a current carrying conductor is directly proportional to the strength of current (), length of element (), sine of the angle (θ) between the element and the line joining element to the point and inversely proportional to the square of distance (r) between the element and the point. db θ db = θ where is the permeability of free space = 4π 10 N/A or Tm/A Comparison between Biot Savart s law and Couloumb s law Both are long ranged. Both are inverse square laws. Principle of super position can be applied to both. Electro static field is produced by a scalar source, charge. Magnetic field is produced by a vector source, current element. Electrostatic field is along the line joining the source charge and the point. Magnetic field is perpendicular to the plane containing the displacement vector and the current element. Biot Savert s law is angle dependent. Coulomb s law is angle independent. Current element, () is a vector quantity. It has the same direction as. Magnitude of current element is the product of current and length of the element. Its unit is ampere metre (Am). Unique Learning Centre, Ulloor, Tvpm. Mob: Page 6

7 Vector form of Biot-Savart s law θ We have, db = Π Multiplying numerator and denominator by r db = db = Π Π θ ( ). This is the vector form of Biot Savart s law. It is clear that db is perpendicular to the plane containing dl and rand is given by Right hand scew rule. Biot Savart s law in terms of current density J = = = = where is volume. On substituting in db = db = Π Π ( ) ( ) Biot Savart s law in terms of charge and velocity dl= = dl = q where is velocity. On substituting in db = db = Π ( ) ( ) The force between two charges and moving with velocities and, F = Π Magnetic field on the axis of a circular loop of current Consider a circular loop of radius R carrying a current. Take a small element on it. The magnetic field at a point P which is at a distance r from it, db = Π ( ) Here the angle between dl and r is 90 0 r = Sin90 = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 7

8 From figure, db = db = Π Π r 2 = x 2 + R 2 db = Π ( ) The direction of db is perpendicular to the plane containing and. r db can be resolved into two components db x (along the axis) and db y (perpendicular to the axis). The y components, due to diametrically opposite elements cancel each other. So the net magnetic field is along the axis. db x = db Cosθ The angle between and is (90-θ). From figure Sin (90-θ) = or Cosθ = Cosθ = ( ) db x = db Cosθ Number of elements in the ring = Magnetic field due to all the elements, = B = B = Vectorially B = Field at the centre of the loop Here x = 0 B = B = ( ) ı Π ( ) ( ) Π π π ( ) ( ) ( ) ( ) ( along the axis) ı where ı is a unit vector along the axis of the circular loop. Magnetic field lines due to a circular loop of current ı Unique Learning Centre, Ulloor, Tvpm. Mob: Page 8

9 Magnetic field at the centre of concentric circular currents (i) B = + (ii) B = Clock rule (Used to find the direction of magnetic field around a current carrying loop) Imagine facing one end of a current carrying loop. If the current at that end is clockwise, that end is south. If it is anti- clockwise, that end is north. Maxwell s right handed screw rule: (Used to find the direction of magnetic field around a current carrying conductor) Imagine the rotation of a right handed screw along the current carrying conductor. If the direction of advancement of its tip represents the direction of current, then the direction of motion of thumb gives the direction of magnetic field. Finding polarity of a current carrying solenoid: It can be found by using right hand grip rule or clock rule. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 9

10 Here thumb gives the direction of magnetic field along the axis of solenoid. Magnetic field due to a long straight conductor carrying current Consider a long conductor carrying a current. Consider a small element of length. P is a point at a distance from it. Magnetic field at P due to is, = Draw AC BP (A) From ACB, Sinθ = AC = θ (1) From the sector PAC, dφ = AC = φ (2) From (1) and (2) θ = φ (3) Equation (A) can be modified as db db From POA, cos φ = Putting in (4) db = I Magnetic field due to the entire wire, = = OR = B = B B B = π cos d = π sin (4) = π sin sin ( ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 10

11 B = π sin sin B = π sin + sin B = π sin + sin If the conductor is infinitely long, = = 90 B = π B = B π = π The direction of this field is obtained from right hand thumb rule. Here the field at P is into the plane of paper. Vectorially, B = π n where n is a unit vector perpendicular to the plane of paper into it. Direction of magnetic field around a current carrying conductor Direction of magnetic field around a current carrying conductor can be obtained from right hand thumb rule. Imagine holding the current carrying conductor in right hand with thumb pointing in the direction of current. Then the curly fingers represent the direction of magnetic field. Ampere s swimming rule: (Used to find the direction of deflection of magnetic needle kept under a current carrying conductor) Imagine a person swimming along the current carrying conductor in the direction of current facing a magnetic needle under it. Then the north pole of the needle will be deflected towards his left hand side. For a current carrying conductor, the magnetic field at a distance d is, B = B Unique Learning Centre, Ulloor, Tvpm. Mob: Page 11

12 Variation of magnetic field with distance for a current carrying conductor for a point in the interior and exterior of the conductor Magnetic field inside a straight pipe carrying current is zero. Magnetic field at a point inside a straight cylinder carrying current B = where R is the radius and r is the distance from axis to the point. Force between two straight parallel conductors carrying current Consider two conductors of length l 1 and l 2 carrying current i 1 and i 2, in the same direction. Let the distance between them be d. F 21 Consider the first conductor. Take a point at a distance d from it (at right side) Magnetic field at the point B 1 = µ π The second conductor is kept there. (into the plane of paper by right hand thumb rule) Force acting on it = θ Here i = i 2, l = l 2, B = B 1, θ = 90 0 Force = i 2 l 2 µ π Force on unit length, F 21 = µ π Sin 90 (towards right by Fleming s left hand rule) Similarly magnetic field at a point d from second conductor (at left side) is B 2 = µ π The first conductor is kept here. Force on it = sinθ Here i = i 1, l = l 1, B = B 2, θ = 90 0 Force = i 1 l 1 µ π (outwards from the plane of paper by right hand thumb rule) sin 900 Unique Learning Centre, Ulloor, Tvpm. Mob: Page 12 l 1 i 1 F 12 d l 2 i 1

13 Force on unit length, F 12 = µ (towards right by Fleming s left hand rule) π F 12 = F 21 Thus if the currents are in the same direction the wires attract. If the currents are in the opposite direction they repel. Definition of ampere When = = = 1A and d = 1m Force on unit length = 2 10 One ampere is that current, which when passed in each of the two parallel infinitely long conductors of negligible cross section placed in vacuum at a distance of 1 m from each other produces between them a force of 2 10 newton per unit length. Magnetic field at a point, half way between two wires carrying current of same magnitude in the same direction = = 0 If currents are in the opposite direction B = Magnetic field around parallel current carrying wires + = 2 Torque acting on a current loop which is kept in a magnetic field Consider a rectangular loop carrying current. It is kept in a magnetic field. The sides are a and b respectively. The angle between perpendicular to the plane of loop and magnetic field is θ. The force acting on arm PQ = θ Here =, =, =, θ = 90 0 Unique Learning Centre, Ulloor, Tvpm. Mob: Page 13

14 Force, = (into the plane of paper and perpendicular by Fleming s left hand rule) The force acting on RS, = θ Here =, =, =,θ = 90 0 Force = ( outwards from the plane of paper and perpendicular by Fleming s left hand rule) The force acting on arms QR and SP = θ = θ These act along the axis of the coil in the opposite directions.hence they cancel each other. The forces on PQ and RS constitute a couple. Moment of couple=torque() = one of the forces X perpendicular distance between their lines of action. But = X Sin θ X = (area of the loop) = θ If there are N turns, = θ = θ where m is magnetic moment. Work done in rotating a coil through an angle from magnetic field direction W = mb(1-cos θ) Direction of magnetic field due to a circular loop of current If the direction of circular current is represented by the curly fingers of right hand then the stretched thumb gives the direction of magnetic field. Ampere s circuital law and its application to find the magnetic field due to a current carrying straight wire, solenoid and toroid Ampere s circuital law Line integral of magnetic field over a closed path in free space is equal to times the total current enclosed by the path.. = Magnetic field due to an infinitely long straight conductor carrying current Consider a long conductor carrying current. P is a point at a distance r from it. Including P we consider a closed path. It is a circle of radius r. This is the Amperian loop. Its plane is perpendicular to the conductor. At P, the angle between and a small element is zero.. = = B = B 2 = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 14

15 Solenoid B = An insulated wire wound in the form of a helix is called solenoid. For a solenoid, turns must be closely wound and diameter of coil is smaller than length. Consider an infinitely long solenoid carrying current. Let be the length. To find the magnetic field on the axis, we take a rectangular Amperian loop.. = B. dl +. + B. dl + B. dl On the arm ab, the angle between and a small element is zero (as the magnetic field is along the axis). B. dl = B dl On the arm bc the angle between and a small element is = Cos = On the arm cd the value of B=0 B. dl = 0 On the arm da the angle between and a small element is B. dl = B dl. = B. dl 90 = 0 = B dl By Ampere s circuital law. = = B dl = = where N is the total number of turns = Or = = But the number of turns in unit length, = Magnetic field due to a toroid Unique Learning Centre, Ulloor, Tvpm. Mob: Page 15

16 Toroid is an endless solenoid in the form of a ring. Consider a toroid of radius r carrying current. Magnetic field has constant magnitude everywhere inside the toroid. But the magnetic field is zero in the open space interior(point P) and exterior point(point Q). On the axis,the magnetic field is along the axis of toroid. We consider three circular Amperian loops. (i)for the open space inside the toroid The amperian loop is taken as 1 and let its radius be. If is the magnetic field, By Ampere s circuital law,. = = As = 0 = 0 Or = 0 (ii)for the exterior of the toroid The Amperian loop is taken as 3 and let its radius be. If is the magnetic field, By Ampere s circuital law,. = = Here the current enclosed is zero as the current entering is cancelled by the current coming out. As = 0 = 0 Or = 0 (iii)for the point inside the toroid The Amperian loop is taken as 2 and let its radius be = r. If is the magnetic field, By Ampere s circuital law,. = = B = where N is the total number of turns B 2 = B = = B = For the Amperian loop, at every point of this loop B either disappears or is perpendicular to the loop or is uniform and tangential to the loop. Limitations of Ampere s circuital law It is not a universal law. It deals with steady currents only. Biot - savart s law and Ampere s circuital law give the same physical result of electric current in two different ways. Both connect magnetic flux density () and current (). Unique Learning Centre, Ulloor, Tvpm. Mob: Page 16

17 Magnetic field at each end of a solenoid = Variation of B with the length of solenoid Magnetic field inside a solenoid is uniform. But field outside is non uniform. When a current is passed through a solenoid, as the currents in the neighbouring turns are parallel, the turns attract each other. Hence the solenoid contracts. A loop of flexible wire having irregular shape and carrying current is placed in a magnetic field. What will be the expected shape of the loop? Circle with plane perpendicular to the field direction. Circle has maximum area and hence maximum number of field lines pass through it. Tension developed in the wire = Magnetic dipole moment of a current loop Consider a current loop kept in a magnetic field. The current in the loop is. The torque on the loop is = -----(1) where A is the area of loop and is the angle between and perpendicular to the plane of loop. But the torque on a magnetic dipole kept in an external field = mb sin (2) Comparing (1) and (2) we get = If there are N turns m = Unit : Am 2 = Magnetic moment of a revolving electron which is the magnetic moment of the loop. Consider an electron revolving round a nucleus in an orbit of radius r. Current produced = But T = = = The magnetic moment associated with this circulating current is = = = Multiplying and dividing by = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 17

18 But = (angular momentum of electron) = = (1) (in to the plane of paper) is called gyro magnetic ratio. But angular momentum = = = Putting in (1) This is the expression for magnetic moment associated with an electron in an orbit. If n = 1 = On putting values = 9.27 x Am 2. This value is called Bohr magnetron. The electron possesses orbital magnetic moment and spin magnetic moment. Variation of magnetic field with distance along the axis of a circular loop of current Moving coil galvanometer Moving coil galvanometer consists of a fine insulated coil which is wound over an aluminum frame. It is free to move within the pole pieces of a cylindrical magnet. At the centre of the coil there is a soft iron cylinder. Soft iron helps in concentrating magnetic field lines and to keep the field radial so that the angle between the perpendicular to the plane of coil and magnetic field is always A pointer is attached to the coil. The coil is connected to a phosphor bronze spring. When current is passed through the coil, it experiences torque and it turns through an angle. = θ = 90 = It stops at a position. Then the restoring torque or anti torque in the spring balances the torque. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 18

19 Restoring torque = Kφ where K is the torsinal constant or spring constant K φ = Or φ = φ is called current sensitivity.its unit is radian/ampere. To increase current sensitivity we have to increase the value of A, B,N and decrease the value of K. Voltage sensitivity = φ = φ Voltage Sensitivity = φ = Figure of merit of galvanometer where R is the resistance of the coil. It is defined as the current which produces a deflection of one scale division in the galvanometer. G = φ = Properties of phosphor bronze It is a good conductor of electricity. It does not get oxidized when current is passed. It is perfectly elastic. It is non- magnetic. Its torsion constant (restoring torque per unit twist) is small. Conversion of galvanometer into ammeter Galvanometer can be converted into ammeter by connecting a low resistance called shunt(low resistance) parallel to it. Let be the main current, i g be the current through galvanometer. The current through shunt = PD across galvanometer = PD across shunt i g R G = ( ) r s r s = Use of shunt resistor It protects galvanometer from excess current It is used to covert galvanometer to ammeter Unique Learning Centre, Ulloor, Tvpm. Mob: Page 19

20 It is used to increase the range of ammeter. Resistance of ammeter is smaller than that of galvanometer. As ammeter has small resistance, it is connected in series in a circuit. Due to its small resistance, the current is not affected. If ammeter is connected parallel, it draws large current due to low resistance and may get damaged. For an ideal ammeter, resistance = 0 Ammeter of low range has more resistance than that of more range. Range of ammeter can be increased but cannot be decreased. If the range of ammeter is to be increased from to, the value of shunt, r s = Conversion of galvanometer into voltmeter Galvanometer can be converted to voltmeter by connecting a high resistance in series with it. Let i g be the current. The same current passes through galvanometer and high resistance. V = i g R G + i g R V i g R G = i g R Or R = Resistance of voltmeter is high. When it is connected parallel, it draws only small current. Hence the P.D across the element is unaffected. Voltmeter of low range has low resistance. If an ammeter of range and resistance is to be converted to a voltmeter of range V then the resistance R to be connected R = - If the range of voltmeter is to be increased from to, the value of high resistance,r = ( ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 20

21 If a voltmeter of range and resistance is to be converted to an ammeter of range then the shunt S to be connected S = ( ) The range of voltmeter can be increased or decreased. If the voltage range of voltmeter is to be increased n times then the value of high resistance, = ( 1) Resistance of an ideal voltmeter is infinity. Hall effect The creation of an electric field ( ) in a metal slab perpendicular to both magnetic field ( ) and current density ( ) when a magnetic field ( ) is applied perpendicular to current density ( ) is called Hall effect. Here or = where is Hall constant. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 21

22 5 MAGNETISM AND MATTER Magnetic and non- magnetic substances Substances which are strongly attracted by a magnet are called magnetic substances. eg: iron, cobalt, nickel etc. Substances which are not attracted by a magnet are called non-magnetic substances. Basic properties of magnets Attractive property: A magnet can attract magnetic materials. Directive property: A freely suspended or pivoted magnet aligns in the north south direction. The pole of the magnet towards geographical north is North Pole. The other pole which is towards geographical south is South Pole. Like poles repel each other and unlike poles attract each other. Magnetic poles always exist in pairs. That is monopole does not exist. Magnetic field The space surrounding a magnet where its effect is felt is called magnetic field. Intensity of magnetic field at a point is the force experienced by a unit North Pole kept at that point. Magnetic flux indicates the total number of magnetic field lines passing through a surface. The number of magnetic field lines passing normally through unit area is called flux density. Uniform magnetic field Magnetic field is uniform, if it has the same magnitude and direction at all points in the region. Magnetic field on the surface of the earth is considered as uniform. It extends up to a height of about 5 times the Unique Learning Centre, Ulloor, Tvpm. Mob: Page 22

23 radius of the earth. The magnitude of earth s magnetic field is of the order of T. The magnetic field of earth changes from place to place on the surface of earth. It changes with time at a given place on the earth. The magnetic field around a bar magnet is non uniform. Representation of uniform magnetic field Uniform magnetic field is represented by parallel equi-distant lines as shown in figure. Geometrical length It is the distance between two ends of a magnet. Magnetic length The poles of a magnet are located not exactly at the ends, but slightly inwards. The distance between these two points is called magnetic length Coulomb s law of magnetism The force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them. If and are the pole strengths of poles separated by a distance d, then F F = = 4 10 henry/metre or ( N/A ) If = = 1 unit and d = 1m then = = 10 N Thus unit magnetic pole is that pole which when placed at a distance of 1m from an equal and like pole will repel with a force of 10 N. Magnetic dipole and dipole moment Unique Learning Centre, Ulloor, Tvpm. Mob: Page 23

24 Two equal and opposite magnetic poles separated by a small distance constitute a magnetic dipole. Dipole moment is the product of pole strength and distance between the poles. It is a vector quantity. Its direction is from south pole to north pole. Dipole moment, = 2. Where 2 is the magnetic length. Unit of dipole moment: or J/T or Nm/T. The natural unit of magnetic moment is Bohr magneton. Pole strength is also called magnetic charge. Unit of pole strength is Am (ampere metre) Or N/T. Magnetic line of force Magnetic line of force is the curve whose tangent at any point gives the direction of magnetic field at that point. It is the path of an independent north pole if it is free to do so. Note: Widely spaced magnetic field lines indicate weak field and closely spaced magnetic field lines indicate strong field. Electric field lines do not exist within a charged conductor. Magnetic field lines exist within the body of the magnet. i.e. Electric field lines are discontinuous, but magnet field lines are continuous closed loops. Magnetite Stone is known as lodestone (leading stone) as it was used by ancient navigators to find direction. Compass used in ships are known as gyrocompass. The strength of a magnet decreases on heating or hammering. Properties of magnetic field lines Magnetic field lines of a magnet form continuous closed loops. The tangent at any point on the field line gives the direction of magnetic field at that point. The larger the number of field lines crossing a unit area, the stronger is the magnetic field. Magnetic field lines do not intersect each other. If they intersect it means that at the point of intersection there are two directions for the field which is impossible. Magnetic field lines prefer to pass through magnetic materials. Torque on a magnetic dipole kept in a magnetic field Consider a magnetic dipole of pole strength and length 2 which is kept in a uniform magnetic field. The axis of the dipole makes an angle with the field. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 24

25 The force on north pole = ( in the direction of ) The force on south pole = ( in the opposite direction of ) These two equal and opposite forces constitute a couple. Moment of couple is called torque. Torque = One of the forces Perpendicular distance between their line of action = AB In, Sin = Or AB = 2 Sin Torque = 2 Sin But 2 =, dipole moment = m B Sin OR = which is the torque acting on the dipole kept in a uniform field. Time period of oscillation of a freely suspended magnetic needle kept in a magnetic field When the magnetic needle is kept in a magnetic field, it experiences a torque. = mb Sin where is the torque and is the angle between the axis of dipole and magnetic field. When the magnetic needle is turned, it oscillates and comes in equilibrium position. In the equilibrium state, I = where is the moment of inertia. ve sign shows that restoring torque is in the opposite direction of deflecting torque. Or If is small, Sin + = This is similar to the equation of SHM = 0 = x = 0 Hence the oscillations of the magnetic needle are simple harmonic. 2 = OR = = 2 = π Or Time Period, = π T = = 2 Squaring, 2 = 4 Strength of field B = 4 which is expression for the time period of oscillation of the magnetic needle. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 25

26 Potential energy of a magnetic dipole which is kept in an external field When a magnetic dipole is kept in an external field, it experiences a torque = mbsin When turned through an angle d, work done dw = d Work done in turning through an angle θ is W = W = = W = [-Cos θ] W = Cos θ This work is stored as potential energy. Potential energy, U = θ When dipole is perpendicular to field, θ = 90 0 U = 0 When θ = 0 0 U = mb (potential energy is minimum). This is the most stable position. When θ = U = U = 1 U = (potential energy is maximum). This is the most unstable position. Magnetic field on the axial line of a bar magnet Consider a bar magnet of pole strength and magnetic length 2. P is a point on the axial line at a distance r from the centre of the magnet. Intensity of magnetic field at P due to north pole is, = () where is a unit vector in the same direction of. Intensity of magnetic field at P due to south pole is, = () Total intensity where is a unit vector in the same direction of. = + () () = () () = () () () () = ( ) = ( ) But 2 = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 26

27 Since, is neglected. = ( ) ( in the same direction of dipole moment ) = OR = Magnetic field on the equatorial line of a bar magnet Consider a bar magnet of pole strength and length 2. P is a point on the equatorial line at a distance r from the centre of the magnet. The distance from each pole to P is. The magnetic intensity at P due to North Pole is = along PQ. The magnetic intensity at P due to South Pole is, = along PS. = = B The field along PQ is resolved into two components : B cos along x direction and B sin along y direction. The field along PS is resolved into two components : B cos along x direction and B sin along Y direction. The Y components cancel each other as they are equal and opposite. The x components add up. The total field at P is B = 2B Cos = 2 But 2 = dipole moment. = From figure, = + = ( + ) / = ( + ) / B = ( ) / (parallel to the axis of magnet) Vectorially = << r is neglected. = ( ) / where is a unit vector in the same direction of dipole moment. ( ) / OR = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 27

28 Note : For a short dipole, = 2 Magnetic field at any point = For axial point = 0 and for equatorial point = 90. Comparison between electrostatics and magnetism Magnetism B m Equatorial field for a short dipole Axial field for a short dipole Torque, x x Gauss s law in magnetism Electrostatics E Surface integral of magnetic field over a closed surface is zero.. = 0 P Or The net magnetic flux though any closed surface is zero Gauss s law in magnetism can be applied to both closed and open surfaces. Differential form of Gauss s theorem + + = 0 Consequences of Gauss s law in magnetism By Gauss s law there is no point where the magnetic field lines start or end. Hence magnetic monopoles do not exist. Magnetic poles exists like unlike pairs of equal strength. The number of magnetic lines of force entering a closed surface is equal to the number of lines of force leaving the surface. Show that a current carrying loop behaves as a magnetic dipole. Obtain the expression for its magnetic moment. Consider a circular loop of current whose radius is r which carries a current. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 28

29 The magnetic field along its axis, B = = A, the area of the loop ( ) / = ( ) / = B = (1) The electric field on the axial line of a dipole is =.(2) On comparing (1) and (2) we find that and have the same distance dependence. Also they have the same direction. This shows that the circular loop acts as a magnetic dipole of magnetic moment. m = A Vectorially =. The result is valid for planar loop of any shape. The direction of is normal to the plane of loop as given by right hand thumb rule. Current carrying solenoid as bar magnet Solenoid can be taken as the combination of circular loops arranged side by side. Each turn of the current carrying solenoid acts as a magnetic dipole. The number of dipoles is equal to the number of turns in the solenoid. The north pole of one loop touches the south of the next. Hence their effects cancel each other. That is there will be poles only at the ends of the solenoid. One end of the solenoid act as north pole( the end with anti clockwise current) and the other end act as south pole ( the end with clockwise current). The length of the solenoid is the distance between the poles. Thus the solenoid is equivalent to a bar magnet. Note : Inside the solenoid the direction of magnetic field is from South to North. Finding the direction of the magnetic dipole moment of a current carrying solenoid The magnetic dipole moment of a current carrying solenoid can be found using right hand moment rule. If the curly fingers of right hand represent the direction of current in the solenoid, then the stretched thumb gives the direction of magnetic moment. Mapping of magnetic field lines of a bar magnet Magnet placed with its north pole pointing geographic north: Figure shows the magnetic lines of force of a bar magnet placed in the magnetic meridian with its north pole pointing the geographical north of the earth. Null point (Neutral point) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 29

30 Null points are the points where the magnetic field due to the magnet is equal and opposite to the horizontal component of earth s magnetic field. The net magnetic field at the null point is zero. A compass needle kept at null point can stay in any direction. In this case (N N), the magnetic field due to the magnet on the equatorial line and the horizontal component of earth s magnetic field are equal. ( ) / = where is the horizontal component of earth s magnetic field. Magnet placed with its north pole pointing geographical south. In this case (N S) the magnetic field due to the magnet on the axial line and the horizontal component of earth s magnetic field are equal. ( ) = 1T = 10 4 G (gauss) Note: gauss is the CGS unit of magnetic field intensity. Magnetizing field When a magnetic material is kept in a magnetic field, it gets magnetic power. The magnetic field which exists in vacuum and can induce magnetism is called magnetizing field. eg: The magnetic field developed inside a current carrying solenoid. Magnetic induction indicates the total magnetic field inside the material. It is the sum of external magnetizing field and the magnetic field due to magnetization of material. The unit of magnetic induction is tesla or weber/m 2 Or Nm -1 A -1 or J A -1 m -2. Magnetizing field intensity (magnetic intensity) The ability of a magnetizing field to magnetize a material medium is expressed as a vector called magnetizing field intensity( represented as ). = Unit: ampere/ metre OR Nm -2 T -1 Or J m -1 wb -1. The CGS unit is oersted. 1 oersted = 80 A/m Dynamo Effect The core of the earth is in molten state. It contains elements like iron, nickel etc. With the axis rotation of the earth, these also rotate. This produces current which is responsible for the magnetic field of the earth. This effect is called dynamo effect. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 30

31 Some important terms Magnetic Meridian b) Geographic meridian Magnetic Meridian: It is the vertical plane containing the given place and magnetic north and south poles. Geographic Meridian: It is the vertical plane containing the given place and geographic north and south poles. Elements of earth s magnetic field Declination: It is the angle between magnetic meridian and geographic meridian. Dip (inclination): It is the angle between the net magnetic field of earth with horizontal. Dip at pole is 90 0.Dip is maximum at equator. Dip at equator is zero. Horizontal Intensity( ): It is the resolved component of earth s magnetic field in the horizontal direction. The value of ( ) is zero at poles. Note: Magnetic latitude λ and dip angle are related as Dip needle λ = 2 It is a compass needle pivoted freely to move in a vertical plane containing the magnetic field of the earth. At the poles the dip needle will point straight down. What is the dip angle at a place where the vertical and horizontal components of earth s field are equal? tan = = 1 i.e. = 450 A magnetic needle orients with its axis vertical at a certain place on the earth. What are the values of (a) horizontal component of earth s field (b) angle of dip? Ans:(a) Zero (b) 90 0 The angle of dip and total magnetic field of earth at a place are and B respectively. What are the horizontal and vertical components of earth s magnetic field at the place? = cos and = sin Magnetization GK's PHYSICS Unique Learning Centre, Ulloor, Tvpm. Mob: Page 31

32 A circulating electron in an atom has a magnetic moment. For a bulk material these moments add up vectorially to give a non-zero magnetic moment. Magnetization (M) of a sample is the ratio of net magnetic moment to volume of the sample. M = Some important relations It is a vector quantity. Unit: ampere/metre. Dimension = AL -1 Magnetic field in the interior of a current carrying solenoid B 0 = 0 If the solenoid has a core, magnetization increases to B = B 0 + B m where B m = 0 M Expression for magnetic intensity (H) We have = 0 ( + ) = Or 0 = 0 ie = - Unit: ampere/metre. Dimension: AL -1 Magnetic susceptibility (χ) Susceptibility χ =. It is the ratio of magnetization to magnetic intensity. It has no unit and no dimension. Some important relations = 0 ( + ) But = H = 0 (H + M) = 0 (1 + ) = 0 (1 + χ) But = 0 r 0 r = 0 (1 + χ) or r = 1 + χ When a magnet is cut into two pieces If a magnet is cut into two along its length, into two equal parts, the pole strength reduces to half. Dipole moment reduces to half. If it is cut into two equal parts perpendicular to length, pole strength remains same, dipole moment reduces tohalf. Lines joining places of same declination are isogonal lines. Lines joining places of zero declination are agonal lines. Lines joining places of same dip are isoclinic lines. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 32

33 Line joining places of zero dip are aclinic lines (magnetic equator). Line joining places of same horizontal intensity is called isodynamic lines. The extent to which magnetic field lines can enter a substance is called magnetic susceptibility. A substance which can be easily magnetized has large value of permeability. Properties of diamagnetic substances Diamagnetic substances are substances which experience weak repulsive force from a magnet When a diamagnetic material is kept is a non-uniform magnetic field, it moves from stronger region to the weaker region of the field. Diamagnetism is independent of temperature. Their relative permeability is less than one. Their susceptibility is small and negative Reason for repulsion from external field Orbiting electron in an atom possesses orbital angular momentum. Resultant magnetic moment in an atom is zero. When external magnetic field is applied, electrons having angular momentum in the same direction of field slow down and opposite direction speed up. Substances develop a net magnetic moment opposite to external field and are repelled from the field. Eg: Bi, Cu, Ag, Au, Pb, Zn, quartz, glass, diamond, water, alcohol, NaCl, hydrogen, helium, argon, nitrogen (at S.T.P), Hg, marble etc. Figure shows the behaviour of a diamagnetic material in an external field. The field lines are repelled or expelled. The perfect diamagnetism exhibited by super conductor is known as Meissner Effect Properties of paramagnetic substances Paramagnetic substances are the substances which are weakly magnetized when placed in an external field. When a para magnetic substance is kept in a non-uniform field, it moves from weaker region to stronger region of the field. The relative permeability of a para magnetic material is greater than one. The susceptibility of a para magnetic material is small and positive. Behaviour of paramagnetic material in an external field Unique Learning Centre, Ulloor, Tvpm. Mob: Page 33

34 For a para magnetic material, individual atoms or molecules possess a permanent dipole moment. Due to random arrangement no net magnetization is found. When kept in a strong external field, the dipoles are aligned and point in the same direction of field. The field lines get concentrated inside the material. The field inside is strengthened. Magnetization is inversely proportional to temperature. M Or M = or M = where C is curie constant. Here is the magnetizing field. χ This law is called Curie s law. = or χ = Example for para magnetic substances: aluminum, sodium, manganese, magnesium, chromium, platinum, tungsten, lithium, niobium, calcium, oxygen (at STP), copper chloride etc. Properties of ferro magnetic substances Ferromagnetic substances are substances which are strongly magnetized when kept in an external field In a non-uniform field, it moves from weaker to stronger region of the field Relative permeability of a ferro magnetic material is very high. Susceptibility of a ferro magnetic material is high and positive. Individual atoms or molecules possess a permanent dipole moment. They inter act with each other and align in a common direction over a macroscopic volume called domain. Each domain is of 1mm and contains nearly atoms. Magnetism varies from domain to domain and bulk magnetism is zero. When external field is applied the domains arrange in the same direction of field. Boundaries of domains of ferromagnetic material are called block walls. eg: Iron, cobalt, nickel, alnico etc. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 34

35 The behaviour of ferro magnetic substance in an external field is shown below. The field lines are highly concentrated. Curie point When a ferro magnetic substance is heated, its magnetic properties decreases. At a particular temperature the ferromagnetic substance becomes para magnetic. This temperature is called Curie point(curie temperature). The curie temperature of iron is about 1043 K. Modified Curie s law for ferro magnetic substances Above Curie point(ie, in para magnetic condition), χ = where C is a constant. Here >.This is known as Curie- Weiss law. ie, The susceptibility of a ferro magnetic substance above its Curie temperature is inversely proportional to the excess temperature above Curie temperature. The earth s core is known to contain iron, but geologists do not regard this as a source of magnetic field. Why? The temperature of core is more than the curie temperature of iron. At this temperature, iron loses its magnetic properties. At the equator, the vertical component of earth s magnetic field is zero. The vertical component of earth s magnetic field at a place is 3 times the horizontal component. What is the dip angle? tan =, tan =, tan = 3 OR = tan -1 3 The angle of dip of a location in southern India is about Would you expect a greater or lesser dip at Britain? More. It is located close to North Pole of earth. Dip angle is about 4 times than in southern India. Difference between hard ferromagnetic substances and soft ferromagnetic substances Consider a ferromagnetic substance which is kept in an external magnetic field. When external field is removed some ferromagnetic substances retain the magnetic power they got. They are called hard ferromagnetic substances. Hard Ferro magnetic material can not be demagnetized easily. Eg: alnico (alloy of aluminum, nickel, cobalt, iron and copper), lode stone Some ferromagnetic substances cannot retain the magnetic power when external field is removed. They are called soft ferromagnetic substances. Soft ferromagnetic material can be demagnetized easily. Eg: soft iron, cobalt, nickel, gadolinium etc. Variation of B with H field Unique Learning Centre, Ulloor, Tvpm. Mob: Page 35

36 Consider a ferromagnetic substance which is kept inside a solenoid. Suppose initially there is no magnetization to the substance. When external field is increased by increasing the current in the solenoid, the magnetic field B in the material increases. It reaches saturation at A. Now the H field is decreased to zero. But the substance retains some magnetism. The point B represents retentivity or remanence. Now the external field is reduced to ve value by reversing the current in the solenoid. At C the magnetic field in the substance becomes zero. This ve value of H is called coercivity. Again H is decreased. Magnetic saturation is again attained (at D). Now H is reversed by decreasing the current in the solenoid. B also increases and again the point A is reached. For a given value of H, the value of B is not unique. The phenomenon of the lagging of magnetization behind magnetizing field is known as hysteresis. The area of hysteresis loop represents the energy dissipated as heat in the specimen when it undergoes a cycle of magnetization. Area of hysteresis loop of hard ferromagnetic material is more. Hysteresis loop for soft iron and steel Graph showing the variation of intensity of magnetisation () versus H field for two materials A and B is given. Identify them? Why B has large for a given field at constant temperature? Unique Learning Centre, Ulloor, Tvpm. Mob: Page 36

37 B A H = is more for B. So B is ferromagnetic and A is paramagnetic. Ferromagnetic material is easily magnetized. So its value is more. Properties of materials suitable for making permanent magnets a)high permeability b)high retentivity c) High coercivity Materials suitable for making permanent magnets Steel, alnico, cobalt steel, ticonal etc. Properties of materials used for making the core of electromagnets a) High permeability b)low retentively Soft iron is a suitable material for making electro magnets. Vibration magnetometer It is an instrument used to compare the magnetic moments of two magnets. It can also be used to determine the horizontal component of earth s magnetic field at a place. In it, a magnet is allowed to oscillate simple harmonically in a magnetic field. Time period of oscillations, T = 2 where is the moment of inertia, m is the magnetic moment and is the horizontal component of earth s magnetic field. If two magnets are used, For the first magnet = 2 For the second magnet = 2 OR = If the magnets are placed as in the figures below, Total moment of inertia = + Total magnetic moment = + Total moment of inertia = + Total magnetic moment = - Unique Learning Centre, Ulloor, Tvpm. Mob: Page 37

38 Time period, T = 2 Time period, T = 2 ( ) Magnetic moment of two identical magnets each of moment inclined at 60 0 with each other ( ) m= cos 60 = = 3 = 3 What is a natural fundamental magnetic dipole? Electron A magnetic needle is made to float on the calm surface of a lake in northern hemi sphere. Will it move towards north? It will come to rest along North South direction If a compass is taken to North Pole of the earth, it moves in horizontal plane. It may rest in any direction. Magnetic field lines of a spinning electron resemble a magnetic dipole. A submarine is an iron shell which shields the compass from the magnetic field of the earth. So the compass is ineffective inside. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 38

39 6 ELECTRO MAGNETIC INDUCTION Electromagnetic induction When the magnetic flux linked with a coil or conductor changes, an emf is developed in it. This phenomenon is known as electromagnetic induction. The emf developed is called induced emf. If the circuit is closed, an induced current passes through it. Experiment to show the production of emf in a coil by electromagnetic induction Consider an insulated coil. Connect its ends to a galvanometer. Introduce north pole of a bar magnet suddenly into it. The galvanometer shows deflection which indicates the production of current in the coil. When the North Pole is withdrawn, the galvanometer shows deflection in the opposite direction. It indicates the production of current in the opposite direction. In both cases emf is developed across the ends of coil.the experiment can be performed by introducing or withdrawing the south pole. In that case, the deflection of galvanometer will be opposite to that inthe first case. In the figure below and are insulated coils.k is a tapping key. What happens when a) The key is pressed? b) The key K is pressed continuously? c) When the key is released? Give reasons. Answer: a) Galvanometer shows a momentary deflection. When the key is pressed a magnetic field is developed around (The magnetic flux increases from zero to Unique Learning Centre, Ulloor, Tvpm. Mob: Page 39

40 maximum). The same flux change is liked with as it is very close to. So an emf is developed and an induced current passes through the circuit. b) Galvanometer shows no deflection. Here the magnetic flux is not changing. No emf and hence no induced current are developed. c) Galvanometer shows deflection in the opposite direction. The change of magnetic flux in this case is in the opposite direction (maximum to zero).hence emf and induced current are developed in the opposite direction. Faraday s law of electromagnetic induction The magnitude of induced emf in a circuit is equal to the rate of change of magnetic flux linked with it. Induced emf, = - ve sign indicates that induced emf opposes magnetic flux change. If there are N turns =. = induced emf = But =.. =. This is called integral form of Faraday s law. Lenz s law The direction of induced emf is given by Lenz s law. The polarity of induced emf is such that it produces current which opposes the magnetic flux change that causes its production. Prove that Lenz s law is in accordance with the law of conservation of energy. Consider an insulated coil, the ends of which are connected to a galvanometer. When North Pole of a bar magnet is introduced into it, the magnetic flux through the coil increases. An induced current is developed in the coil which opposes the increase of magnetic flux. This current makes the near end of the coil (end near the north) north (anti clockwise current). Thus it is difficult to introduce the North Pole. Some mechanical work has to be done in overcoming repulsion. The energy spent for this appears as electrical energy in the coil. When the North Pole is withdrawn, the magnetic flux through the coil decreases. An induced current is developed in the coil which opposes the decrease of magnetic flux. This current makes the near end of coil south (clock wise current). Some mechanical work has to be done in overcoming attraction. The energy spent for this appears as electrical energy in the coil. Thus Lenz s law obeys law of conservation of energy. Induced emf in a conductor when moved in a magnetic field Consider a straight rectangular conductor PQRS moving in a uniform magnetic field. Let the direction of field be perpendicular to the plane of paper and into it. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 40

41 Let the arm PQ be movable. Its length is. RQ = The magnetic flux enclosed by PQRS is = B.A = B A [ as the angle between B and A is zero] But A = = PQ is moved with a velocity v as in figure. When it is moved, the magnetic flux decreases as the area decreases. Induced emf in it, = = is taken as = x = If R the resistance of the conductor, Note : =. Induced current, = = The arm carrying current is in the magnetic field B. It experiences a force F = ( ) = 90 F = (in the opposite direction of v) F = = To move the arm, power required, P = Force velocity P = P = The mechanical energy spent appears as electrical energy. Direction of induced emf in a conductor when moved in a direction perpendicular to a magnetic field It can be found by using Fleming s Right Hand Rule: Stretch the forefinger, middle finger and thumb of right hand in mutually perpendicular directions. If fore finger represents the direction of magnetic field, thumb represents the direction of motion of conductor then the middle finger represents the direction of induced emf or current (also called generator rule). Unique Learning Centre, Ulloor, Tvpm. Mob: Page 41

42 Induced emf developed in a rod when rotated in a magnetic field Consider a rod of length rotated in a uniform magnetic field B (into the plane paper and perpendicular) with an angular velocity. Angle swept = Area swept in time t A = = = Flux linked = BA cos = BA cos 0 = BA = B = Induced emf = = Induced current = = = = Induced emf between the centre and rim of a disc radius R, rotating with a constant angular speed in a magnetic field B along the axis of rotation of the disc = When a conductor is moved parallel to the magnetic field, no emf is developed in it. Why? Here the magnetic flux linked with the conductor does not change. So no emf is developed. What is the magnitude of induced current in the loop of radius r if the straight wire PQ carries (1) steady current (2) increased current from P to Q? Unique Learning Centre, Ulloor, Tvpm. Mob: Page 42

43 Zero as the current is steady The magnetic field above the wire is perpendicular to the plane of paper and outwards by right hand thumb rule. The induced current in the loop must oppose it. Hence the current must be clock-wise. When a light is switched off, spark is produced in the switch. Why? When the circuit is switched off, the current in the circuit decreases suddenly. Large induced emf is developed between the contacts of the switch which tries to maintain current in the circuit. Air in between the contacts loses the insulation property and spark is developed. Eddy currents PHYSICS When the magnetic flux linked with a bulk piece of conductor changes, a circulating current is produced in it. This current is called eddy current (Foucault current). Eddy currents are like eddies (like whirling air or water). Resistance of a metal is low. So eddy currents are large. Eddy currents heat metal. It is disadvantageous for transformers, induction coil etc. Eddy currents can be minimized by making slots in the metal. (When slots are made, the available area decreases). In a transformer, to reduce eddy currents the core is laminated by varnishing it. A copper plate is suspended between pole pieces of a magnet. What happens when it is oscillated? What is the change when slots are made in the copper plate. Explain? It comes to rest in a short time. It is because eddy currents are developed in it during motion. During motion the magnetic flux linked with the plate changes as it moves in and out of the region between poles. Eddy current obeys Lenz s law. The tendency of current is to decrease the speed of motion. In the next case, when slots are made, the available area decrease. So eddy current decreases. The plate will not come to rest suddenly. A ferrous bar falling vertically through the hollow region of a thick cylindrical shell made of copper experiences retarding force. What is special about the bar? Bar experiences retarding force means it is a magnet. When it falls down, eddy currents are developed in the shell which opposes motion. Magnetic braking system in trains In some trains there are electromagnets. When these are activated, strong magnetic field is created over the rails(under the compartments). Eddy current are developed in the rails as the magnetic flux is changing due to the motion of train. Eddy current obeys Lenz s law. The tendency of this eddy current is to stop the train. furnacesgk's Induction Unique Learning Centre, Ulloor, Tvpm. Mob: Page 43

44 In an induction furnace a strong magnetic field is created by passing high frequency alternating current through a coil. The metal to melted is kept at its centre. Eddy currents are developed in it. Eddy currents heat the metal to extremely high temperatures and it melts. Induction cooker In an induction cooker a strong magnetic field is created by passing alternating current through a coil. The steel vessel is kept in this field. Eddy current s are developed in the vessel. Eddy currents heat the vessel to high temperatures and food gets cooked. An aluminium disc is placed over an electro magnet. What happens when the supply is switched on? Why? The disc will be thrown up. When the supply is switched on, the magnetic flux linked with the disc changes. Eddy current are developed in it. So the disc is slightly magnetized. If the upper end of the coil is north then the eddy current makes the bottom part of the disc north. Due to the repulsion between the poles, the disc flies up. Working of speedometer In a speedometer, a magnet rotates inside an aluminium cylinder, according to the speed of the vehicle. Eddy currents are developed in the aluminium cylinder which rotates it. A pointer attached to it indicates the speed of the vehicle. Dead beat galvanometer In a galvanometer a fine insulated coil is wound over an aluminium frame. The coil is between the cylindrical magnetic poles. When current is passed through the coil magnetic flux linked with the aluminium frame changes. Eddy currents are developed in it which opposes the motion of the coil (as it obeys Lenz s law). This type of stopping or damping is called electromagnetic damping. Electric power meter In a power meter an aluminium disc is kept near a coil. When ac is passed through the coil, the magnetic flux linked with the disc changes. Eddy currents are developed in it and it rotates. Induction motor In an induction motor there is stationary coil and a rotor that can be rotated freely. When alternating current is passed through the stator coil, large eddy currents are developed in the rotor and it rotates. Disadvantages of eddy currents Eddy currents develop heat in core of devices like transformers which causes loss of electrical energy. Heat developed in core may cause burning of the insulation of coil. Eddy currents may cause unwanted damping effect. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 44

45 Inductance An emf is developed in a coil by the flux change produced by the same coil or flux change produced by another coil in its vicinity. If is the flux and is the induced current, If there are N turns, N = constant The constant in this situation is called inductance. It is a scalar quantity. Its dimension is. Its SI unit is henry (H). Mutual induction The phenomenon of production of emf in a coil due the change in the magnetic flux in the neighboring coil caused due to a varying current is called mutual induction. Consider two co- axial insulated coils of area A and length. The number of turns in the inner coil is and the number of turns in the outer coil in. The inner coil is and outer coil is. When current is set up in,a magnetic flux is set up in. Then or =.(1) is the mutual inductance of. and. From or = = A..(2) where and represent the number of turns in unit length of (1) and (2) = A = (A) Similarly when current is set up in a magnetic flux is set up in. Then =.(3) where is the mutual inductance of = = (4) = or = (B) = = M (say). where M is the mutual inductance or coefficient of mutual induction. This is known as the reciprocity theorem of mutual inductance. PHYSICS Self induction GK's Unique Learning Centre, Ulloor, Tvpm. Mob: Page 45

46 The phenomenon of production of emf in a coil due to the change in the magnetic flux in the same coil due to a varying current is known as self induction. Consider a coil of N turns carrying. Magnetic flux linked with it N N = N = = or = = OR = The emf produced as a result of self induction is called back emf. Thus when the magnetic flux linked with a coil changes, an emf is developed in it in the opposite direction of applied emf. This emf is called back emf. It opposes the growth and decay of current in the circuit. Self inductance is known as inertia of electromagnetic induction because it opposes the growth and decay of current. Mutual inductance of two concentric coils of different radii M = 2 (where << ) Mutual inductance of two co axial coils of different radii M = ( ) / R > r Factors affecting mutual inductance Number of turns of coils Common area of cross section of coils Relative separation and orientation of two coils Permeability of core material. where x is the distance between the centres of two coils Coefficient of coupling between two coils of self inductance k = For tight coupling k = 1. For loose coupling k = 0. Factors affecting self inductance Number of turns of coil Permeability of core material Area of cross section of the coil. Generally k is less than one. Self inductance increases if air core is replaced by soft iron core. Self inductance of series and parallel combination of coils Unique Learning Centre, Ulloor, Tvpm. Mob: Page 46

47 Series L = + Parallel L = (Here the assumption is that the magnetic flux due to one coil is not linked with other) Induced emf is non conservative = (1) But Lenz s law = - From (1) and (2). = - Non conservative (2) 0 Induced emf is never greater than emf applied. 1 henry = 10 ab henry or 10 emu of inductance. 1 henry = 1 Wb/ampere Induced emf is also called virtual emf. For a bird sitting on a high voltage line, the wings are repelled due to induction which makes it flies away. 1 Wb = 10 maxwell Dimensional formula of magnetic flux Induced emf is called back emf as it opposes the cause which produces it. Induced emf has no direction of its own. Why? Induced emf depends on increase or decrease of magnetic flux linked with the coil. Pure inductor must have negligible resistance. Non inductive winding This type of winding is used to reduce self induction. The coil is wound as shown. The magnetic field in the neighboring turns cancel each other as the currents are in the opposite direction. But the resistance is doubled as the length is doubled. In the circuit given, when the key is closed, the lamp glows dimly. When the key is released, the lamp glows brilliantly for a moment. Why? Unique Learning Centre, Ulloor, Tvpm. Mob: Page 47

48 When the key is closed, the inductor opposes the growth of current and the bulb glows dimly. When the key is released, an induced emf is developed in the coil which opposes the decay of current. Large current flows through the bulb for a moment and it glows brightly. Explain what happens when K is closed in the circuit given? glows suddenly. glows slowly. When current passes through L, an emf is developed which opposes the growth of current in it. But no such emf is developed in the resistor R. Energy stored in an inductor When the magnetic flux linked with a coil changes, a back emf is developed in it. In establishing current, work has to be done against back emf. Rate of work is = where is the magnitude of induced emf and is the current. = = ( ) = () = = = Total work done increasing the current from0 to = W = W = This work is stored as magnetic potential energy Energy stored = A bar magnet is allowed to fall down along the axis of a solenoid. Compute the acceleration of the magnet with respect to acceleration due to gravity. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 48

49 When the bar magnet freely falls towards the solenoid, magnetic flux linked with the coil increases. An emf is induced in it. This emf obeys Lenz s law. Hence there is opposition to the movement of magnet. Net force on the magnet = mg F where F is the opposing force and mg is its weight. Acceleration of the magnet = = g - which is less than g. When the magnet is well inside the solenoid there is no change for the magnetic flux linked with the coil. No emf is developed. So it falls under gravity. Acceleration of magnet = g When it comes out of the solenoid the magnetic flux linked with the solenoid decreases. An emf is developed in it. The motion of magnet is opposed. Acceleration of magnet = g - A rectangular loop of wire is pulled to right from a long straight wire through which a steady current is passed upwards. Does the induced current in the loop flow in clockwise or anti clockwise direction? The magnetic field at the right side of the wire is into the plane of paper. When the loop is moved, the magnetic flux linked with it decreases. The direction of induced current in the loop must be in such a way that it must oppose the decrease of magnetic flux. So it must be clock-wise( by right hand thumb rule). A closed loop PQRS is moved into a uniform magnetic field perpendicular to the plane of paper into it. State the direction of induced current in the loop. When the loop is moved, the magnetic flux linked with it increases. The direction of induced current in the loop is such that it opposes the increase of magnetic flux(by Lenz s law). So it will be SRQPS( by right hand thumb rule). Working of an ac generator It consists of an insulated coil which is free to rotate on a rotor shaft. The coil is kept between two magnetic poles. When the coil is rotated, the magnetic flux linked with it changes. An emf is developed in it. The ends of the coil are connected to external circuit by means of slip ring and carbon brushes. Thus we get current in the Unique Learning Centre, Ulloor, Tvpm. Mob: Page 49

50 external circuit. The magnetic flux linked with the coil = B.A = BA where is the angle between B and area vector A. Here = 0 If there are N turns = BAN Induced emf in the coil = = (BAN) = If is the angular speed of the coil, = = = = Maximum emf = BAN sin 90 = BAN = or = When the plane of coil is perpendicular to magnetic field = = 0 Induced emf = 0 = 0 When the coil turns through 90 from the initial position, = = 90 Induced emf = 90 = (maximum emf) When the coil turns through 180 from the initial position, = = 180 Induced emf = 180 = 0 When the coil turns through 270 from the initial position, = = 270 Induced emf = 270 Unique Learning Centre, Ulloor, Tvpm. Mob: Page 50

51 = 1 = (Maximum emf in the opposite direction) When the coil turns through 360 from the initial position, = = 360 Induced emf = 360 = 0 = 0 It can be shown graphically as in figure. As the direction of emf is continuously changing, we get ac in the external circuit. The frequency of ac generated in India is 50Hz. Electro magnetic shielding When a magnetic field is directed towards a conducting sheet, eddy currents are produced in the sheet. The change in the magnetic field is partially detected at a point on the other side of the sheet. This is called electro magnetic shielding. If the conductivity of the sheet is more, the shielding is more effective Unique Learning Centre, Ulloor, Tvpm. Mob: Page 51

52 7 ALTERNATING CURRENT Alternating current is the current that reverses its direction periodically many times a second. It is generated using ac generators. The ac voltage generated during one half rotation of armature of generator and the next half rotation together forms one cycle of ac. The number of cycles produced in one second is called frequency of ac. The average of instantaneous value of ac emf over a full cycle is zero. Thus we take the rms value of emf. It is the square root of mean of square of instantaneous values of emf over a full cycle. The average value of ac over half cycle = current. = = = 2 = Ac cannot be used for electroplating, electrolysis etc. Ac circuit containing resistor only Figure shows a resistor connected to an ac source. Instantaneous value of voltage V = (1) On applying Kirchhoff s loop rule = Or = = where is the maximum value(peak value) of = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 52

53 = = sin (2) From (1) and (2) we find that current and voltage are in same phase. It is shown graphically and in a phase diagram. Power of an ac circuit containing resistor only For an ac circuit containing resistor only, Instantaneous value of voltage = Instantaneous value of current = Instantaneous value of power = = = Average value of power over a cycle is = < > But = [since = ] = < ( ) > = < ( 1-cos2 ) > = < < cos2 > = 0 = But = cos2 > = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 53

54 *Only for reference < cos2 > = cos2 = = sin 2 = [sin 2 sin 0] = [sin 2 2 sin 0] = = = = 0 Ac circuit containing inductor only [sin 2 2 sin 0 [sin 4 sin 0] 0 0 Figure shows an inductor connected to an ac source. Instantaneous value of voltage V = sin (1) On applying Kirchhoff s loop rule V = Putting in (1) = sin = sin Integrating = sin dt = = sin ( - ) [since cos = sin ( ) = [ sin ( ) ] [since sin ( A) =sin A Unique Learning Centre, Ulloor, Tvpm. Mob: Page 54

55 OR = = sin ( ) is known as inductive reactance ( ) sin ( ) is taken as = ( ) (2) From (1) and (2) we can find that current lags behind voltage by a phase of. It is shown graphically and in a phase diagram. Power of an ac circuit containing inductor only For ac circuit containing inductor only, The instantaneous value of voltage V = sin The instantaneous value of current = ( ) The instantaneous value of power = = sin ( ) = sin ( ) Average value of power, = < sin ( ) > = < sin (sin cos cos sin) > = < sin cos - sin cos sin > But c = 0 and sin =1 = < sin 0 sin cos > Unique Learning Centre, Ulloor, Tvpm. Mob: Page 55

56 = < sin cos > = < = < 2 sin cos > sin 2 > = < sin 2 > But < sin 2 > = 0 = *Only for reference < sin 2 > = sin 2 = = = = = = Ac circuit containing capacitor only = cos 2 = cos 2 2πϑT cos 0 [cos 2 2 T cos 0] [cos 2 T cos 0] [cos 4 - cos 0 ] [1-1] =0 Figure shows a capacitor connected to an ac voltage. Instantaneous value of voltage V = sin (1) From Kirchhoff s loop rule V = cos 2 cos 0] Putting in(1) = sin OR q =C sin Instantaneous value of current = = (C sin ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 56

57 = C sin = C cos = cos = sin ( + ) [since cos = sin ( + ) where = is called capacitive reactance sin ( + ) = sin ( + ) is taken as = sin( + ) (2) From (1) and (2) we find that the current is ahead of voltage by a phase of phase diagram. Power of an ac circuit containing capacitor only For an ac circuit containing capacitor only, The instantaneous value of voltage = sin The instantaneous value of current = sin( + ) The instantaneous value of power But sin =1 and cos =0 = = sin sin( + ) = sin [sin cos + cos sin ] = sin cos + sin cos sin. It is shown graphically and in a P = sin 0 + sin cos 1 P = sin cos Power over a cycle = < sin cos > = < 2sin cos > Unique Learning Centre, Ulloor, Tvpm. Mob: Page 57

58 = < = sin 2 > < sin 2 > But < sin 2 > = 0 = 0 Series LCR circuit connected to ac Figure shows a series LCR circuit connected to ac. If V is the instantaneous value of voltage, from Kirchhoff s loop rule, = (A) The current is the same in the three components as they are connected series. The instantaneous value of voltage = sin (1) sin = But = + + Sin = ( ) + Sin = which is similar to the equation of damped S.H.M. + The solution is q = ( + ) (B) = = cos ( + ), = cos ( + ) (2) Putting these in equation (B) = sin ( + ) sin = sin ( + ) + cos ( + ) + ( + ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 58

59 Sin = [R cos ( + ) sin ( + ) + sin ( + )] Sin = [R cos ( + ) +( -L ) sin ( + ) sin = [ cos ( + ) + ( )sin ( + ) ] Multiplying and dividing by Z, sin = Z [ cos ( + ) + ( ) sin ( + ) ] Put = cos and ( ) = sin Sin = Z [ ( + ) + sin ( + )] ( ) = + Sin = Zcos( + ) Using cos = sin( + ) We can write, sin = Z sin + + Comparing both sides, = Z (3) and = (4) From (3) From (4) Putting these in (2) = = = = co( + ) ] ( ) = ( ) = co[ - ( + )] = sin ( + ) (5) Impedance of the circuit, = + ( ) From (1) and (5) we can see that the current and voltage differ by a phase of. This can be represented in a phase diagram. Sin = ( ) Cos = tan = ( ) OR = ( ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 59

60 Phase diagram treatment for LCR circuit Consider a series LCR circuit. The same current passes through all the components. Let be the current. The voltage across resistor ( ) is in same phase with current. The voltage across coil ( ) is ahead of current by. The voltage across capacitor ( ) is behind current by. If > the resultant of and is - V = + ( ) V = + ( ) V = + ( ) tan = = Z = + ( ) ( ) Impedance, = + ( ) = ( ( ) ) = ( ) = ( ) The current and voltage differ by a phase of. Power of a series LCR circuit For an LCR circuit, the instantaneous value of voltage V = sin (1) Instantaneous value of current = sin ( + ) (2) Instantaneous value of power P = sin sin ( + ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 60

61 P = sin [sin cos +cos sin ] P = sin cos + sin cos sin P = sin cos + 2 sin cos sin P = sin cos + sin2 sin The average value of power over a cycle = < sin cos + sin 2 sin > <sin > = = cos <sin > + and < sin 2 > = 0 = cos = cos = cos sin < sin 2 > = cos where cos is called power factor. Ac circuit containing resistor and capacitor in series Consider an ac circuit containing a resistor and a capacitor in series. The instantaneous value of current is = sin (1) The voltage across resistor is in same phase with current. The voltage across capacitor is behind current by The resultant voltage is V. It is behind current by a phase of. This can be represented in a phase diagram. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 61

62 Resultant voltage = sin ( ) (2) tan = = = ( Instantaneous value of power = = sin sin ( ) Average value of power over a cycle The average value of sin over a cycle is. ) = sin [sin cos - cos sin ] = sin cos - sin cos sin = sin cos - = sin cos - The average value of sin2 over a full cycle is zero. Series LR circuit connected to ac Unique Learning Centre, Ulloor, Tvpm. Mob: Page 62 = < sin cos - = < cos <sin > = cos = Consider an ac circuit containing inductor and resistor in series. The instantaneous value of current is = sin (1) The voltage across resistor is in same phase with current. The voltage across inductor is ahead of current by phase of. The resultant voltage is ahead of current by a phase of. The resultant voltage = sin ( + ) This can be represented in a phase diagram. 2 sin cos sin sin 2 sin sin 2 sin > sin < sin 2 > cos = cos

63 Instantaneous value of power, P = sin sin ( + ) LC circuit = sin [sin cos +cos sin ] = sin cos + Unique Learning Centre, Ulloor, Tvpm. Mob: Page 63 = < sin cos > + < = cos < sin > + = cos + sin 0 = cos = cos = cos 2 sin cos sin sin < sin 2 > sin < sin 2 > Consider a series LC circuit connected to ac.the same current passes through both components. Let be the current. The voltage across inductor is ahead of current by. The voltage across capacitor is behind current by. The resultant voltage = (if > ) The resultant voltage is behind the current by If >, the resultant voltage is - The resultant voltage is ahead of current by.. Resultant voltage V = or - V = or V = -

64 V = ( ) or V =( - ) = Z = or = Z = - For LC circuit If = sin V = sin ( + ) or sin ( ) = cos = cos = 0 For purely resistive circuit = 0 cos = 1 = cos = ie, power dissipation is maximum. For purely inductive or capacitive circuit = 90 = rad cos = 0 = cos = 0 No of power is dissipated. For LCR circuit, = Resonance in LCR circuit ( ) For an LCR circuit = + ( ) where = and = If is changed, and change. At a particular value of, become equal to. Then Z = + = R The circuit will have minimum impedance. At resonance, the series LCR circuit behaves like a resistive circuit. At resonance, the current in the circuit is maximum. This particular value of is called resonant frequency.this condition is called resonance. At resonance = = = or L = or = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 64

65 Q factor (Quality factor) sharpness of resonance For an LCR circuit, the current in = At resonance, L = The current in the circuit will be maximum. = ( ) Maximum value of current at resonance = If frequency is other than the current will be less than. We choose a frequency for which current is Power dissipated by the circuit becomes half. We select two frequencies and equaidistant from.. = + and = The quantity is called sharpness of resonance (Q factor). For, the current = - = ( + ) ( - ) = 2 is called band width. = = OR 2 = + ( Squaring 2 = + ( = ( OR R = R R We have = R ) = ( + )L = (1 + ) ( ( ) ( OR = = (1 + ) ( ) ) ) ) R OR = L = (1 + ) (1 + ) As ) Unique Learning Centre, Ulloor, Tvpm. Mob: Page 65

66 R = (1 + ) L (1 ) R R R = + L L = + L + = 2 Quality factor (sharpness of resonance) = Sharpness of resonance (Q factor) is the ratio of resonance frequency to band width. Sharpness of resonance (Q factor) = = = = (1) Thus sharpness of resonance (Q factor) is the ratio of inductive reactance to resistance at resonance. = OR = Putting in [1] Q factor = = OR L = = OR = Thus sharpness of resonance (Q factor) is the ratio of capacitive reactance to resistance at resonance. LC oscillators The figure shows a fully charged capacitor connected to an inductor. Applying Kirchhoff s loop rule to it. = (1) The current in the circuit is = sign shows that as increases, q decreases. (1) by L, Putting the value of we get, = 0 + = 0 + = 0 or + = 0 Unique Learning Centre, Ulloor, Tvpm. Mob: Page 66

67 This is of the form + = 0 which is the equation of S.H.M. The charges undergo SHM in the LC circuit. The solution is q = cos ( t + ) Initially q = cos t The charge is maximum when t = 0 (initial time). Now energy is stored in capacitor only. E = CV2 = C = Energy stored in inductor = 0, Total energy = + 0 = When t = the capacitor gets fully discharged. The current in the circuit is maximum. Energy stored in inductor = 2 At t = the capacitor is charged by the inductor in the opposite direction. Energy stored in inductor = 0. Energy stored in capacitor =. Total energy = Again at t = + 0 = the capacitor gets fully discharged. There is energy only in the inductor. At t = T again the capacitor is fully charged. The process can be briefly explained as given below: The current passes from +ve plate to ve plate of capacitor through inductor. When capacitor is discharged fully the current gets maximum strength. The magnetic field around the inductor has maximum strength. Now the current stops. The magnetic field disappears. Due to the sudden change of magnetic flux, an emf is developed in the coil. This charges the capacitor in the opposite direction. Now the current flows through the coil in the opposite direction. When the capacitor is fully discharged, the current becomes maximum. Now the current stops. The magnetic field of full strength around the coil disappears. An emf is developed in it. Itcharges the capacitor in the opposite direction. Thus energy is continuously shifted between the capacitor and coil. But due to the resistance of the circuit, the oscillations are damped. The frequency of oscillations, = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 67

68 Transformers Transformers work with the principle of mutual induction. There are two insulated coils in it primary and secondary. Primary is wound over a soft iron piece. Secondary is wound over primary. AC is passed through primary. A magnetic field having variations is developed around it. The secondary is also in this field. So an emf is developed in it. If the number of turns in secondary is more, emf is more.if the number of turns is minimum, emf is minimum. Transformer equation =, N s -Number of turns in secondary, N p Number of turns in primary, V s secondary voltage, V p primary voltage. Difference between step up & step down transformers Step up It increases ac voltage Primary contains lesser number of turns Secondary contains more turns Primary wire is thick Secondary wires are thin Step down It decreases ac voltage Primary contains more turns Secondary contains lesser number of turns Primary is thin Secondary wires are thick. For a transformer we assume that power input = power output. increases with increase of frequency as = L = L 2 or α For dc, frequency, = 0 = = = infinity For dc, frequency, = 0 = L = L 2 = 0 Unique Learning Centre, Ulloor, Tvpm. Mob: Page 68

69 The unit of impedance is ohm. For an LCR circuit, R is the resistance offered by resistor, the total resistance of C and L is. These are taken as the sides of a triangle ABC. Then AC represents impedance (Z). This triangle can be called as impedance triangle. The reciprocal of reactance is called susceptance. The reciprocal of impedance is called admittance. Series LCR circuit allows maximum current at resonance. So it can tune a particular frequency or filter unwanted frequencies. In a radio receiver, we use an LCR circuit. When a number of frequencies from different stations strike the antenna, the circuit allows only that frequency which is its own resonance frequency. Thus a particular station can be selected. This is the working of a ratio tuner. The capacitor in this circuit will be a variable capacitor. Series LCR circuits are also called acceptor circuits or tuned circuits or filter circuits. In a parallel LC circuit resonance is at =. In this condition it does not allow any current. Metal detector Metal detector works with the principle of resonance of ac circuits. It consists of a coil of many turns connected to a capacitor. This circuit will be in resonance. When a person walks with a metal in his pocket through this coil, the impedance of the circuit changes. So the current in the circuit changes which is detected using buzzer which produces sound. Variation of Z,, and R with frequency of applied ac Parallel LC circuits do not allow current at resonance. Thus they act as rejecter circuits. Wattless and wattful currents P = cos. In a circuit where and differ by a phase of 90 o, is resolved into two components: Unique Learning Centre, Ulloor, Tvpm. Mob: Page 69

70 cos and sin. The power due to cos = ( cos ) Cos 0 = cos Also the power due to sin = ( sin ) Cos 90 = 0 Power due to sin is zero. Such a component of current due to which the power is zero is called wattless current (idle current). The power due to the other component cos is called wattful current. In LC circuit, the resistance of the circuit plays the role of friction and decreases the amplitude of oscillations. Damped LC oscillations are shown below. As the energy of LC circuit is dissipated as heat, it becomes warmer with time. For a transformer, we assume that the magnetic flux is the same for both primary and secondary. The induced emf in secondary = Voltage in primary Dividing (1) by (2) = = K (1) = (2) = which is called transformer ratio or turns ratio. K is less than 1 for step down transformer K > 1 for step up transformer. Efficiency of transformer η = = For an ideal transformer η = 100 % Energy losses in a transformer Copper loss Primary and secondary are copper winding. These wires will have some resistance. So energy is wasted as heat. Flux leakage Sometimes the entire flux linked with primary will not be associated with secondary. This causes loss. Iron loss The core is made of iron. Eddy currents are developed in it. The core gets heated. This causes energy loss (eddy current loss). When the core undergoes a cycle of magnetization, it gets heated causing energy loss called hysteresis loss. During the working of a transformer, the core vibrates. This produces a humming noise. Thus energy is lost as mechanical energy. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 70

71 Large scale production of electricity is in ac form. Why? When the electricity is passed through wires, a major part is lost ac heat. Heat can be reduced by reducing current. When voltage is increased using step up transformer, current decreases. Step-up transformer works with ac only. At some stages of distribution, the voltage is reduced using step- down transformers. They also work with ac only. Ideal resistor has only ohmic resistance. But real resistor processes some inductance and capacitance also. Real inductor will have some ohmic resistance in addition to inductance. A real capacitor is like a series LCR circuit. Choke coil is an inductor used to control current in an ac circuit without any loss. It reduces voltage by producing back emf. Series resonance circuits used for voltage amplification. Parallel resonance circuits are used for current amplification. In three phase transmission the voltage between two phases is 415 V and that and that between phase and neutral is i.e. 240 V. In India it is 400 V and 230 V. Core of a transformer is made of a magnetic material. Why? As the material is of high permeability, the magnetic flux with the coil will be completely linked with it. The magnetic flux linked with primary will be completely linked with secondary also. Core of transformer is made of a material of small hysteresis loop. Why? During the working, the core undergoes magnetization so many times. So it gets heated. If hysteresis loop is small, heat will be minimum. Transformer will not work with dc. Why? DC has no variation. When it is passed through the primary, the magnetic flux linked with it does not change. So no emf is developed in secondary. In a step- up transformer, primary wires are thick. So the current in primary is more. Ac measuring devices have non linear scale. Why? Ac measuring devices work on the heating effect of current. Heat is proportional to the square of current. So divisions are not linear. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 71

72 Calculate the rms value of ac from the figure below. = ( ) = 2 A The instantaneous value of ac is E = 300 sin 314 t. What is the rms voltage of the source? = = = V At high frequency of ac, capacitor behaviors as conductor. Why? = = When f 0. So it has no resistance to ac. It acts like a conductor. The best method of reducing voltage in an ac circuit is by using a choke as there is no loss of power. Does a transformer contradict the principle of energy conservation? No We cannot define rms value of current in terms of chemical effect of current as ac cannot be used to cause chemical effect. Heating effect is most suitable for measuring current as heat is proportional to square of current and is independent of the direction of current. Radio frequency chokes are air cored. Why? = = L 2. For radio frequency (high frequency) will be large and the choke will block the entire current. If there is core L is also large. Iron loss in a transformer can be reduced by laminating the core. AC is more dangerous than dc. The peak value of ac is more than that of dc. E.g. for 220 V ac and ac and 220 V dc, the peak value of dc is 220 V. But peak value of dc is 220 V. But peak value of ac is 2 = = For a series LCR circuit where R is a bulb frequency f is doubled how the values of C and L should be changed so that the glow of bulb is unchanged? = = 2 = =, to get same, when f is doubled, L must reduce to half. To get the same the capacitance C must become half. Induction coil is used to generate high ac voltage from suitable dc voltage. In a dc dynamo, split ring commutator helps to get dc. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 72

73 8 ELECTRO MAGNETIC WAVES Inconsistency in Ampere s circuital law Consider the process of charging a capacitor by passing a time dependent current i(t). Consider a point P outside the capacitor. To find the magnetic field at a point outside the capacitor, we consider a closed circular path of radius r as in figure. The plane of the circle is perpendicular to the current. According to Ampere s circuital law. = i(t) On the circle, the angle between and the element is zero. = i(t) or = i(t) B 2 = i(t) B = () To find the magnetic field we can consider a pot shaped surface with the same circle as its mouth and whose base is between the plates of capacitor. If we consider a closed path between plates, no current passes through it.. = 0 If we proceed, we get the value of magnetic field as zero. Thus we get two different values for the magnetic field at the same point. This is a contradiction. To overcome the difficulty Maxwell suggested the existence of a new current called displacement current. The electric field between the plates E = E = But = The electric flux in the region between plates = EA = A = Differentiating OR = = Unique Learning Centre, Ulloor, Tvpm. Mob: Page 73

74 = = where is the displacement current. Thus displacement current appears in situations when electric field and electric flux change with time. The total current is the sum of conduction current and displacement current. Outside the plates of capacitor there is no displacement current. But in between the plates of capacitor there is displacement current only and no conduction current. Thus ampere s circuital law is modified as. = i c + OR. = + This equation called Ampere s Maxwell equation. Maxwell s equations. = Q (Gauss s law in electricity). = 0 (Gauss s law in magnetism). = ( Faraday s law). = i c + ( Ampere s Maxwell law) Generation of electromagnetic waves Accelerated charges radiate electromagnetic waves. where is the conduction current and is the displacement current. Hertz succeeded in producing electromagnetic waves for the first time. Hertz s Experiment to produce electro magnetic waves The experimental set up consists of two large metal plates which are connected to metal spheres as in figure. There is a small air gap between the spheres. The high voltage from an induction coil is connected to the plates. The plates act as capacitor and the induction coil provides inductance. Thus it is an LC circuit. Hence there is continuous transfer of energy between the inductor and the capacitor. A spark is seen between the spheres which shows the oscillations of charge which sends out an electromagnetic wave. The frequency of em waves in Hertz s experiment = The detector of em waves consists of a metal ring whose ends are connected to metal spheres. Hertz produced em waves of wave length about 6 metre. Representation of the electric and magnetic fields of electromagnetic wave in which electric field is along X direction, magnetic field is in y direction and direction of propagation of wave is along z direction Unique Learning Centre, Ulloor, Tvpm. Mob: Page 74

75 E x = E o sin (kz t) By =B o sin (kz t) where k is the propagation constant k = λ, E o and B o are the peak value of electric field and magnetic fields. Marconi succeeded in sending electromagnetic waves over a large distance. Properties of em waves em wave requires no medium for propagation. They possess a speed of m/s in vacuum. Em wave is a combination of electric & magnetic fields. In an em wave, electric and magnetic fields are in mutually perpendicular planes. Velocity of an em wave in a medium = If and B o are the electric and magnetic fields then = c They carry both energy and momentum. When an electromagnetic wave strikes an absorbing surface, momentum delivered to the surface P = where U is the energy and c is the velocity of em wave. For a perfectly reflecting surface, the momentum imparted by em wave = em waves can be reflected, diffracted, refracted. It also shows interference. em waves can be polarized. em waves are not charged. So they are not deflected by electric and magnetic fields. Electromagnetic waves are transverse waves. em wave transport both energy and momentum. Speed of an em wave = Electromagnetic spectrum The orderly arrangement of em waves (in the order of the wave length or frequency) is called em spectrum. Unique Learning Centre, Ulloor, Tvpm. Mob: Page 75