ELECTRO MAGNETIC INDUCTION AND ALTERNATING CURRENT

Transcription

1 PEMI ELECTO MAGNETIC INDUCTION AND ALTENATING CUENT C C Magnetic Flux Like electric flux, magnetic flux, B, through a surface then B = B. S ds is defined as and it represents total lines of induction crossing through a given surface S. Magnetic Induction and Faraday s Laws B S B.dS. If B is uniform If the magnetic flux through a circuit or closed loop changes, an emf and a current are induced in the circuit. This phenomenon is known as electromagnetic induction and the law which governs this phenomenon is known as Faraday s Law. This law states that the magnitude of induced emf in a circuit is equal to the time d rate of change of the magnetic flux. Mathematically, e. As B.A BA cos. Hence if there is any change in magnetic field (B) or area (A) or orientation () then there is induced emf. If some situation, more than one of these may contribute in induced emf, in this case magnitude of induced emf is written as e d db da d (BA cos ) (A cos ) (Bcos ) BAsin This induced emf creates an induced current in the circuit whose magnitude is given as induced emf e I. Also the charge flown = net resistance of circuit Practice Problems :. A circular coil (constant radius) of total length L having number of turns N is rotated about the diameter in a uniform magnetic field B with an angular velocity. Initially the magnetic field is perpendicular to the plane of the coil. The maximum value of the emf induced in it is. BL N NBL BL 4N NBL 4. A thin circular ring of area A is held perpendicular to a uniform magnetic field of induction B. A small cut is made in the ring and a galvanometer is connected across the ends such that the total resistance of the circuit is. When the ring is suddenly squeezed to zero area, the charge flowing through the galvanometer is AB/ AB/ AB 4 AB 3 [Answers : () c () b] C3 Lenz s Law The direction of induced emf is governed by Lenz s Law. This law states that an induced emf is always in the direction that opposes the change of magnetic flux that induced it. Incorporating this law into Faraday s d Law, the induced emf is given by e. The negative sign indicates that the induced emf opposes the change of the flux. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

2 Note the Lenz s Law is based on conservation of energy principle. Practice Problems : PEMI. In the figure the flux through the loop perpendicular to the plane of the coil and directed into the paper is varying according to the relation = 6t + 7t + where is in milliweber and t is in seconds. Choose the correct statement : At time t = s, the current flowing through is ma from left to right At time t = s, the current flowing through is ma from right to left The current through is always increasing linearly both and are correct. A rectangular coil (having resistance per unit length /3 /m) of turns and size. m.5 m is placed perpendicular to a magnetic field of. T. If the field drops to.5 T in.5 s then the magnitude of average induced current is 4mA the total charge flown in the coil is 5µC the total charge flown in the coil isindependent of time during which the field will change both and are correct 3. A solenoid has turns wound over a length of.3 m. Its cross-sectional area is. m. Around its central section a coil of 3 turns is wound. If an initial current of A flowing in the solenoid is reversed in.5 s, the emf induced in the coil will be 6. 4 V 6. V V 4.8 V [Answers : () d () c (3) d] C4 Motional Electromotive Force If a conductor with length L moves with speed v in a uniform magnetic field with magnitude B, and if the length and velocity are both perpendicular to the field, the induced emf is e = vbl. More general, when a conductor moves in a magnitude field B, the induced emf in the direction is given by e (v B).d l Practice Problems :. An electric potential difference will be induced between the ends of the conductor shown in the diagram when it moves in the direction a b Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

3 PEMI 3 P Q L M. A conducting square loop ABCD of side L and resistance moves in its plane with a uniform velocity v perpendicular to one of its sides. A magnetic induction B, constant in time and space, pointing perpendicular and into the plane of the loop exists everywhere, then The current induced in the loop is zero There is no induced emf in the rod BC and AD There is an induced emf BLv in each rod AB and CD All the above statements are correct [Answers : () d () d] C5 Induded Electric Field : When an emf is induced by a changing magnetic flux through a stationary closed path, there is an induced electric fleld E of non-electrostatic origin such that d E.dl Properties of Induced Electric Field. It is not a Coulomb field. B. The lines of induced field form closed loop. Therefore, it is called a circuital field or vortex field. 3. This field is nonconservative and cannot be associated with a potential. Practice Problems :. Consider a cylindrical space of radius in which a time varying magnetic field is confined. Find the dependence of induced electric field on the distance r from the centre inside the space and outside the space? [Answers : () inside E is directly proportional to r and outside it is inversely proportional to r] C6 Self inductance and Inductors Any circuit that carries a varying current will have an emf induced in it by the variation in its own magnetic field. Such an emf is called a self-induced emf. Self-induced emf s can occur in any circuit, since there will always be some magnetic flux through the closed loop of a current-carrying circuit. But the effect is greatly enhanced if the circuit contains a coil with N turns of wire. As a result of the current i, there is an average magnetic flux B, through each turn of the coil. Here we defined the self inductance L of the circuit as follows N L I B Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

4 The SI unit of inductance is the henry (H). Self inductance of the solenoid PEMI 4 The inductance per unit length near the middle of a long solenoid of cross-sectional area A and n turns per unit length is Self induced emf L µ n l A The self-induced emf, using Faraday s law, is given by Practice Problems : di e L. The current in a coil changes from to A in.5 s. If the induced emf is 8 V, the self-inductance of the coil is H.5 H.5 H H. A torodial solenoid with an air core has an average radius of 5 cm, area of cross-section cm and turns. Ignoring the field variation across the cross-section of the toroid, the self-inductance of the toroid is 4.6 mh 6.9 mh.3 mh 9. mh 3. A coil is wound on a frame of rectangular cross-section. If all the linear dimensions of the frame are increased by a factor and the number of turns per unit length of the coil remains the same, selfinductance of the coil increases by a factor of [Answers : () d () c (3) b] C6 Energy Stored in an Inductor If an inductor L carries a current i. the inductor s magnetic field stores an energy given by C7 L Circuits : Applying Kirchoff s voltage law across an inductor. U Li If the direction of assumed current coincides with the direction of motion, the voltage across the inductor di falls and is given by L. If the direction of assumed current is opposite to the diretion of motion the voltage across the inductor rises di and is given by L. Growth of Current in L circuit : Let us connect a coil of self-induction L with a resistance across a cell of emf E as shown in figure. If the switch S is thrown in contact at t =, current i in the circuit tends to grow. Hence an emf is induced across the coil in such a direction as to oppose this current. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

5 By Kirchoff s voltage law, we have PEMI 5 E i L di di E i L i di E i t L [log(e i)] i = t i(t) = L E e t L Here L is known as time constant of the circuit. The current grown in the circuit exponentially as shown in figure. Note the following points :. At t =, i =, we can say at t =, the inductor behaves like a breaking wire.. In steady state : At t, E i, we can say at t, the inductor behaves like a connecting wire. 3. The rate at which the source or battery will supply energy = Ei, rate at which the energy is dissipated in resistor = i and the rate at which the energy stored in the inductor = energy Ei i di i L. i L di. From conservation of Decay of current in L circuit : At t =, the current passing through the inductor is I and it is connected across a resistor as shown in figure : di di Using KVL, i L I L Practice Problems : i di I L t I I e. In the following circuit initially there is no current through the inductor. Find the current passing through the battery at any time t. Also find the current through the battery at t = and t =. t / L Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

6 PEMI 6. A solenoid has an inductance of 53 mh and a resistance of.37. If it is connected to a battery, how long will the current take to reach half its final equilibrium value? 3. A solenoid having an inductance of 6.3 µh is connected in series with a. k resistor. If a 4. V battery is switched across the pair, how long will it take for the current through the resistor to reach 8.% of its final value? What is the current through the resistor at time t =. L? 4. At time t =, a 45. V potential difference is suddenly applied to a coil with L = 5. mh and = 8. At what rate is the current increasing at t =. ms? [Answers : (). s (3) 8.45 ns; 7.37 ma (4). A/s] C8 C9 Energy Density of a Magnetic Field If B is the magnitude of a magnetic field at any point (in an inductor or anywhere else), the density of stored magnetic energy at that point is Mutual Induction B u B. µ When a changing current i in one circuit causes a changing magnetic flux in a second circuit, an emf e is induced in the second circuit; likewise, a changing current i in the second circuit induced an emf e in the first circuit. This is called mutual induction. di di e M and e M The constant M, called the mutual inductance, depends on the geometry of the two coils and on the material between them. If the circuits are coils of wire with N and N turns, respectively, the mutual inductance can be expressed in terms of the average flux B through each turn of coil that is caused by the current i in coil or in terms of the average flux B through each turn of coil that is caused by the current i in coil : C N M i B N i B The SI unit of mutual inductance is the henry, abbreviated H. Equivalent units are H = Wb/A = V.s/A =.s. Mutual inductance of two solenoids one surrounding the other is given by µ n p n s Al where n p and n s are number of terms per unit length for primary and secondary coils and A is the cross-sectional area of primary coil and l is the length of the primary coil. LC Circuit An L-C circuit, which contains inductance L and capacitance C, undergoes electrical oscillations with angular frequency : LC Such a circuit is analogous to a mechanical harmonic oscillator, with inductance L analogous to mass m, the reciprocal of capacitance /C to force constant k, charge q to displacement x, and current i to velocity v. Practice Problems :. A capacitor of capacitance µ F is charged upto V and then connected across an ideal inductor of mh. Choose the correct statement : The angular frequency of LC oscillation is 4 rad/s At any moment total energy is 5µJ The current in the circuit changes with time sinusoidally All are correct Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

7 PEMI 7. A capacitor of µ F initially charged to V is connected across an ideal inductor of. mh. The maximum current in the circuit is.5 A A.5 A A [Answers : () d () b) C Back EMF in D.C. Motor : A motor is the reverse of generator it converts electrical energy into mechanical energy. When currents is passed through a coil placed in a magnetic field, it rotates. As the coil rotates, the magnetic flux linked with changes, giving rise to an induced emf. This emf opposes the applied emf () and is, therefore, called back emf (e). If is the resistance of the coil, the current through it is given e by I. Practice Problems :. In a dc motor, if E is the applied emf and e is the back emf, then the efficiency is E e E e E E e E e E [Answers : () b] C C3 C4 Eddy Currents When a metallic body is moved in a magnetic field in such a way that the flux through it changes or is placed in a changing magnetic field, induced currents circulate throughout the volume of the body. These are called eddy currents. Alternating Current An alternator or ac source produces an emf that varies sinusoidally with time. Production of A.C. Production of A.C. is based on Faraday s law of electromagnetic induction. Suppose a coil of N turns, and area A is rotated in a uniform magnetic field B with angular velocity. As the coil rotates, the flux through it changes and therefore an emf is induced in it, given by = sin t where = NBA. A sinusoidal voltage or current can be represented by a phasor, a vector that rotates counterclockwise with constant angular velocity equal to the angular frequency of the sinusoidal quantity. Its projection on the horizontal axis at any instant represent the instantaneous value of the quantity. Average and root mean square value of a.c. For a sinusoidal current the average and rms (root-mean-square) currents are related to the current amplitude I by I av I.637I, I Irms. In the same way, the rms value of the snusoidal voltage is related to the voltage amplitude V by C5 V rms V The voltage v in an ac circuit is represented by v = v sint and current in a.c. circuit is represented by i = i sin(t + ) where is the phase angle between the current and voltage. A.C. Circuit Pure resistive a.c. circuit The voltage across a resistor is in phase with the current, and the voltage and current amplitude are related by V = I Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

8 Pure inductive circuit PEMI 8 The voltage across an inductor L leads the current by 9, the voltage and current amplitude are related by V L = IX L, where X L = L is the inductive reactance of the inductor. Pure capacitive circuit The voltage across a capacitor C lags the current by 9 ; the voltage and current amplitudes are related by V C = IX C, where X C = /C is the capacitive reactance of the capacitor. LC series circuit In an ac circuit the voltage and current amplitudes are related by V = IZ, where Z is the impedance of the circuit. In an L-C- series circuit, Z (XL XC ) ( L (/ C)], and the phase angle of the voltage relative to the current is Practice Problems : L / C tan. A 4 electric heater is connected to V, 5 Hz main supply. The peak value of the electric current flowing in the circuit is approximately.5 A 5. A 7 A A. An alternating voltage V = sin t, where V in volt and t seconds, is connected to a series combination of µf capacitor and k resistor through an ac ammeter. The reading of the ammeter will be ma ma ma ma 3. Choose the correct statement : the current leads the voltage in phase if an ac source is connected across a capacitor the current lags behind the voltage in phase if an ac source is connected across an inductor the current and voltage are in same phase if an ac source is connected across a resistor. all are correct [Answers : () c () b (3) d] C6 Power in A.C. circuit The average power input P av to an ac circuit is P av VIcos V rms I rms cos where is the phase angle of voltage with respect to current. The quantity cos is called the power factor. Practice Problems :. If a current I = I sin (t /) flows in a circuit across which an alternating potential E = E sin t has been applied, then the power consumed in the circuit depends on E I both none. In circuit, an alternating current of A flows for minutes. In another similar circuit, a direct current of A flows for the same time. If the heat produced in circuit is X then the heat produced in circuit is.5 X.5 X X X Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

9 PEMI 9 3. A sinusodal alternating current flows through a resistor. If the peak current is I p, then the power dissipated is I p 4 I I p I p 4. The impendence of a circuit consists of 3 resistance and 4 reactance. The power factor of the circuit is [Answers : () d () c (3) b (4) b] C7 esonance in LC Circuit In an L-C- series circuit the current becomes maximum (for a given voltage amplitude) and the impedance becomes minimum at an angular frequency = /(LC) / called the resonance angular frequency. This phenomenon is called resonance. At resonance the voltage and current are in phase, and the impedance Z is equal to the resistance. Practice Problems :. In an LC series circuit, the capacitance is changed from C to 4C. For the same resonant frequency, the inductance should be changed from L to L L/ L/4 4L [Answers : () c] C8 C9 Quality Factor L The Quality factor of an LC series circuit is defined as Q where is the resonance angular frequency. It is an indicator of the sharpness of the current peak higher the value of Q, sharper is the peak. Transformer A transformer converts a low aleternating voltage to a high voltage and vice-versa. It is based on the principle of mutual induction. It consists of two coils wound on a soft iron core. The primary coil is connected to an a.c. source.the secondary coil is connected to the load which may be a resistor or any other electrical device. If the primary resistance is zero, then E p is equal to the applied voltage. Further, if there is no flux leakage, i.e., the same flux is linked with each turn of both the primary and secondary coils, then it can be shown that Es N s. E p N p If N s > N p, then E s > E p and the transformer is called a step-up transformer. If N s < N p, then E s < E p and the transformer is called a step-down transformer. For an ideal transformer, Input power = Output power E p I p = E s I s p s Es Ns. E N In actual transformers, there is some power loss. The main sources of power loss are :. I loss due to Joule heat in copper windings.. Heating produced due to Eddy currents in the iron core. This is reduced by using laminated core. 3. Hysteresis loss due to repeated magnetisation of the iron core. 4. Loss due to flux leakage. When all the losses are minimized, the efficiency of the transformer becomes very high (9-99%). Practice Problems :. In a step-down tranformer the input voltage is kv and the output voltage is 55 V. The ratio of the number of turns in the secondary to that in the primary is : : : 4 4 :. In a noiseless transformer an alternating current of A is flowing in the primary coil. The number of turns in the primary and secondary coils are and respectively. The value of the current in the secondary coil is.8 A.4 A 5 A A [Answers : () c () d] I I p p Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

10 PEMI SINGLE COECT CHOICE TYPE. In the following figure there is a magnetic f i e l d of. Tesla along the positive x-axis. The magnetic flux through the face BCDG is.8mwb.mwb.8mwb. A copper disc of radius is rotated about its centre with n revolutions per second in a uniform magnetic field B. Choose the incorrect statement : If the field is in the plane of the disc then the induced emf between the centre and the edge of the disc is zero If the field is in the plane of the disc then the induced emf between the centre and the edge of the disc is non-zero If the field is perpendicular to the disc then the induced emf between the centre and the edge of the disc is Bn both and are correct 3. A player with 3 metre long iron rod runs towards east with a speed of 3 km/hr. Horizontal component of earth s magnetic field is 4 5 Wb/m. If he runs with the rod in horizontal and vertical positions, then the potential difference induced between the two ends of the rod in the two cases will be zero in vertical position, 3 V in horizontal position 3 V in vertical position, zero in horizontal position zero in both positions 3 V in both positions 4. A square metal loop of side cm and resistance ohm is moved with a constant velocity partly inside a uniform magnetic field of Wb/m, directed into the paper, as shown in the figure. The loop is connected to a network of five resistors each of value 3. If a steady current of ma flows in the loop, then the speed of the loop is.5 cm/s cm/s cm/s 4 cm/s 5. Two inductors, each of inductance L, are connected in series then effective self inductance (L eff ) is given by L eff = L L L eff 3L L eff 4L L L eff 4L 6. A rectangular loop with a sliding connector of length l is located in a uniform magnetic field perpendicular to the loop plane as shown in figure. The magnetic induction is equal to B. The connector has an electric resistance, the sides AB and CD have resistance and respectively. Neglecting the self inductance of the loop, find the current flowing in the connector during its motion with a constant velocity. BlV BlV 3BlV 4BlV Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

11 7. An emf of 5 V is applied in a circuit containing 5 H inductance and resistance. The ratio of the currents at time t = and t = s is e / e / e e e e 8. A rectangular loop of sides 8 cm and cm is lying in a uniform magnetic field of magnitude.5 T with its plane normal to the field. The field is now gradually reduced at the rate of. T/s. If the resistance of the loop is.6, then the power dissipated by the loop as heat is 6.4 W 3. W W 3. 5 W 9. A torodial solenoid with an air core has an average radius of 5 cm, area of cross-section cm and turns. Ignoring the field variation across the cross-section of the toroid, the self-inductance of the toroid is 4.6 mh 6.9 mh.3 mh 9. mh. A coil is wound on a frame of rectangular cross-section. If all the linear dimensions of the frame are increased by a factor and the number of turns per unit length of the coil remains the same, self-inductance of the coil increases by a factor of In the given circuit is a resistor, L is an inductor and B and B are two bulbs. If the switch S is turned off A; 5 A; 5 A A PEMI A; 5 A; A tranformer is used to light a 4 W, 4 V bulb from a 4 V A.C. mains. The current in the main cable is.7 A. The efficiency of the transformer is % 83.3% 6.7% 36.% Two conducting rings of radii r and r move in opposite directions with velocities v and v respectively on a conducting surface S. There is a uniform magnetic field of magnitude B perpendicular to the plane of the rings. The potential difference between the highest points of the two rings is zero rvb 4rvB 8rvB 5. A magnet is moved with a high speed towards a coil at rest. Due to this, the induced emf, the induced current and the induced charge in the coil are E, I and Q respectively. If the speed of the magnet is doubled, the incorrect statement is A A both B and B die out promptly both B and B die out with some delay B dies out promptly but B with some delay B dies out promptly but B with some delay. Two resistors of and and an ideal inductor of H are connected to a V battery as shown. The key K is inserted at time t =. The initial (at t = ) and final (at t = ) currents through the battery are The induced current become I The induced emf becomes E The induced charge remains same The induced charge is Q Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

12 6. The number of turns of primary and secondary coils of a transformer are 5 and respectively and the mutual inductance of the transformes is 5 H. If the number of turns in the primary and secondary are made and 5 respectively, then the mutual inductance of the transformer will be 6.5 H.5 H 5 H 5 H 7. In an LC circuit, choose the correct statement current and voltage are always in phase if LC current lags behind the voltage if / LC current leads the voltage if / LC All are correct 8. A small square loop of wire of side l is placed inside a large square loop of wire of side L (L >> l). The loops are coplanar and their centres coincide. The mutual inductance of the system is µ µ l L L l 4 µ µ l L l 9. Two circular loops of radii a and b (b >> a) are placed coaxially a distance r ( r >> b) apart. The mutual inductance between the loops is µ a b /(r 3 ) µ a b /(r 3 ) 3µ a b /(r 3 ) 4µ a b /(r 3 ). A magnetic flux through a stationary loop with a resistance varies during the time interval as = at ( t). The inductance of the loop is to be neglected. The amount of heat generated in the loop during that time a 3 / / a 3 / /3 a 3 / /4 a 3 /. A current I = 3.36( + t) A increase at steady rate in a long straight wire. A small circular loop of radius 3 m has its plane parallel to the wire and is placed at a distance of m from the wire. The resistance of the loop is The magnitude of the induced current in the loop is 4 A 8 A A 6 A L PEMI. An ac source of angular frequency is fed across a resistor and a capacitor C in series. The current registered is I. If now the frequency of source is changed to /3 (but maintaining the same voltage), the current in the circuit is found to be halved. The ratio of reactance to resistance at the original frequency A km long telegraph wire has capacity of.4 µf/km. If it carries an alternating current of frequency 5 khz, the value of an inductance required to be connected in series so that the impedance is minimum..36 mh.8 mh.9 mh.3 mh 4. A 75 hertz, V source is connected to a resistance of ohm, an inductance of.83 henry and a capacitance of microfarad all in series. The time in which the resistance (thermal capacity J/ C) will get heated by C..8 min. 3.8 min. 4.8 min. 5.8 min. 5. An LC series circuit with resistance is connected to an ac source of V and angular frequency 3 rad/s. When only the capacitance is removed, the current lags behind the voltage by 6. When only the inductance is removed, the current leads the voltage by 6. The power dissipated in the LC circuit is W 4 W 5 W 6 W Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

13 PEMI 3 EXCECISE BASED ON NEW PATTEN COMPEHENSION TYPE Comprehension- A ohm coil of mean area 5 cm and having turns is held perpendicular to a uniform field of.4 gauss. The coil is turned through 8 in (/)s.. The change in flux is 4 mwb 6 mwb 8 mwb mwb. The average induced emf is mv mv 3 mv 4 mv 3. Average induced current is ma 4 ma 6 ma 8 ma 4. The total induced charge is µc µc 3 µc 4 µc Comprehension- An induction furnace uses electromagnetic induction to set up eddy currents in a conductor, thereby heating the conductor. Commercial units operate at frequencies ranging from 6 Hz to MHz and deliver powers from a few watts to several magawatts. Induction heating can be used for welding in a vacuum chamber, to avoid oxidation or contamination of the metal. At high frequencies, induced currents appear only near the suface of the conductor this is the skin effect. By creating an induced current for a short time at an appropriate high frequency, a sample can be heated down to a controlled depth. For example, the surface of a farm tiller can be tempered to make it hard and brittle for effective cutting while keeping the metal s interior soft and ductile to resist breakage. To explore induction heating. Consider a flat conducting disk of radius, thickness b and resistivity. A magnetic field B max cos t is applied perpendicular to the disk. Assume that the frequency is so low that the skin effect is not important. Assume the eddy currents flow in circles concentric with the disk. 5. The maximum power delivered to the disk is B max. b B max. b 6. The average power delivered to the disk is B max. b B max. b 3 B max. b B max. b 4 7. By what factor does the power change when the amplitude of the field doubles? remains same two times becomes half four times Comprehension-3 An air-core toroidal solenoid with cross-section area A and mean radius r is closely wound with N turns of wire. The wire is carrying a current i. The field of an idealized toroidal solenoid is confined completely to the space enclosed by the windings. Assume that the magnetic field inside the solenoid is uniform across cross-section that is, neglect the variation of magnetic field with distance from the toroidal axis. 8. The magnetic field inside the solenoid is µ NI r µ NI r µ NI 4r µ NI r 9. Suppose N = turns, A = 5. cm and r =. m then its self inductance is µh µh 3 µh 4 µh. If the current in the toroidal solenoid increases uniformly from zero to 6. amp in 3. µs. The value of self induced emf is 8 V 6 V 4 V V. The magnetic energy density depends on radius r as r n. The value of n is B max. b 3 B max. b 4 Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

14 Comprehension-4 Superconducting power transmission The use of superconductors has been proposed for power transmission lines. A single coaxial cable could carry. 3 MW (the output of a large power plant) at kv, dc, over a distance of km without loss. An inner wire of radius. cm, made from the superconductor Nb 3 Sn, carries the current I in one direction. A surrounding superconducting cylinder, of radius 5. cm, would carry the return current I.. In such a system, the magnetic field at the surface of the inner conductor is 5. mt 4. mt 3. mt. mt 3. In such a system, the magnetic field at the inner surface of the outer conductor is 5. mt 4. mt 3. mt. mt 4. The energy that would be stored in the space between the conductors in a km superconducting line is.9 MJ.9 MJ 3.9 MJ 4.9 MJ 5. The pressure exerted on the outer conductor is 8 Pa 8 Pa 38 Pa 48 Pa Comprehension-5 In a circuit shown in the figure, switch S is closed at time t =. Thereafter, the constant current source, by varying its emf, maintains a constant current i out of its upper terminal. 6. The time constant of the circuit is L/ zero L/ none 7. The current through the inductor as a function of time is given by i( e t/l ) i( e t/l ) i( e t/l ) i( e t/l ) 8. The current through the resistor equals the current through the inductor at time L L l n ln L l n none Comprehension-6 Choke Coil PEMI 4 A choke coil is an electrical instrument used for controlling current in an a.c. circuit. Choke coil consists of an inductor with very small resistance. Choke coils are used with fluorescent mercury-tube fittings in houses. Let the inductance of the inductor is L and resistance is. Let the voltage applied is V = V sin t. 9. The rms current through the choke coil is ( V L V V V L ) L. The power consumed by the ideal choke coil is zero very low very high none. The power consumed by the choke coil is zero very low very high none. Iron cored chokes are used for reducing low frequency a.c. high frequency a.c. all types of frequencies none 3. Air cored chokes are used for reducing low frequency a.c. high frequency a.c. all types of frequencies none 4. In place of choke coil we can use to reduce the a.c. current resistor only capacitor only Comprehension-7 D.C. motor both can be used neither can be used A D.C. motor converts direct current energy from a battery into mechanical energy. It is based on : torque will act on a current carrying coil placed in the magnetic field. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

15 PEMI 5 We know that, when a coil will be rotated in a magnetic field then there will be the induced emf. This is known as back emf. (C) The voltage across P in () 7.7 volt (D) The voltage across Q in (S) 9.76 volt MULTIPLE COECT CHOICE TYPE. Let the resistance of the coil is, voltage of D.C. source is V and back emf is E. 5. The maximum current is V/ E/ V E 6. The current at any time is V/ E/ V E 7. The efficiency of the motor is E V V E V E V E 8. A motor having an amature of resistance. ohm operates on V mains. At its full speed, it developes a back e.m.f. of V. V E V E The current when the motor is switched on is amp The current when the motor is at full speed is 5 amp The efficiency of the motor is 95.5 % All the above MATIX-MATCH TYPE Matching- A box P and a coil Q are connected in series with an ac source of variable frequency. The emf of source is constant at V. Box P contains a capacitance of µf in series with a resistance of 3. Coil Q has a self-inductance 4.9 mh and a resistance of 68 in series. The frequency is adjusted so that the maximum current flows in P and Q. Column - A Column - B (A) The impedance of P at (P) 77 this frequency in ohm (B) The impedance of Q at (Q) 97.6 this frequency in ohm. A small magnet M is allowed to fall through a fixed horizontal conducting ring. Let g be the acceleration due to gravity. The acceleration of M will be < g when it is above and moving towards > g when it is above and moving towards < g when it is below and moving away from > g when it is below and moving away from A square loop ABCD of edge a moves to the right with a velocity v, parallel to AB. There is a uniform magnetic field of magnitude B, directed into the paper, in the region between PQ and S only. I, II and III are three positions of the loop. the emf induced in the loop has magnitude Bav in all three positions. The induced emf is zero in position II. The induced emf is anticlockwise in position I. The induced emf is clockwise in position III. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

16 PEMI 6 for the second coil at the same instant are I, V and W respectively. Then : 3. I I 4 I 4 I A conducting disc of radius r spins about its axis with an angular velocity. There is a uniform magnetic field of magnitude B perpendicular to the plane of the disc. C is the centre of the ring. No emf is induced in the disc. The potential difference between C and W 4 W V V 7. An inductance L, resistance, battery B and switch S are connected in series. Voltmeters V L and V are connected across L and respectively. When S is closed 4 the rim is Br. 4. C is at a higher potential than the rim. Current flows between C and the rim. A flat coil, C, of n turns, area A and resistance is placed in a uniform magnetic field of magnitude B. The plane of the coil is initially perpendicular to B. If the coil is rotated by an angle about the axis XY, charge of amount Q flows through it. If = 9, If = 8, BAn Q If = 8, Q = If = 36, Q = BAn Q 5. The SI unit of inductance, the henry, can be written as weber/ampere volt second/ampere joule/ampere ohm second 6. Two different coils have self-inductance L = 8 mh, L = mh. The current in one coil is increased at a constant rate. The current in the second coil is also increased at the same constant rate. At a certain instant of time, the power given to the two coils is the same. At that time the current, the induced voltage and the energy stored in the first coil are I, V and W respectively. Corresponding values The initial reading in V L will be greater than in V The initial reading in V L will be lesser than V The initial readings in V L and V will be the same The reading in V L will decrease as time increases while that in V will increase to a maximum value 8. If L, Q, represent inductance, charge and resistance respectively then the units of Q/L will be that of current Q 3 /L will be that of power QL/ will be that of current Q 3 /L will be that of power 9. An AC voltage of angular frequency is applied to a circuit which consists of an inductor of inductance L and a capacitor of capacitance C in parallel. Then across the inductance current is maximum when = /LC voltage is maximum when = /LC current is minimum when = /LC voltage is minimum when = /LC. A capacitor is charged to a potential of V. It is connected with an inductor through a switch S. The switch is closed at time t =. Which of the following statements are correct Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

17 the maximum current in the circuit is V C L potential across capacitor becomes zero for the first time at time t LC energy stored in the inductor at time t LC is CV 4 maximum energy stored in the inductor is CV Assertion-eason Type Each question contains STATEMENT- (Assertion) and STATEMENT- (eason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. (A) (B) (C) (D) Statement- is True, Statement- is True; Statement- is a correct explanation for Statement- Statement- is True, Statement- is True; Statement- is NOT a correct explanation for Statement- Statement- is True, Statement- is False Statement- is False, Statement- is True. STATEMENT- : A cylindrical magnet is placed near a circular coil. If the magnet is rotated about its own axis, no current is induced in the coil. STATEMENT- : There is no change in magnetic flux through the loop. STATEMENT- : Inserting an iron core in a coil increases its self-inductance. STATEMENT- : The self-inductance of the coil depends on the relative permeability. 3. STATEMENT- : Electric field lines produced by time varying magnetic field is closed curves. STATEMENT- : Electric field produced by time varying magnetic field is non-conservative. 4. STATEMENT- : A transformer works on a.c. only and not on d.c. STATEMENT- : In case of d.c., flux will be constant and so no emf will be induced in the secondary. PEMI 7 5. STATEMENT- : If an aluminium plate is moved rapidly through the region between the poles of an electromagnet, it experiences a strong retarding force. However, if slots are cut into it, the force is greatly diminished. STATEMENT- : It is due to the eddy current flowing through the aluminium plate in larger amount, when the slots are not made. 6. STATEMENT- : A metal coil is kept stationary in a non-uniform magnetic field. An emf is induced in the coil. STATEMENT- : emf will be induced only when the flux will change with time. 7. STATEMENT- : The figure shows an inductor L and a resistor connected in parallel to a battery through a switch. The resistance of is the same as that of the coil that makes L. Two identical bulbs B and B are put in series with L and respectively. When S is closed B lights up earlier than B. STATEMENT- : The bulbs will be equally bright after some time when the steady state is reached. 8. STATEMENT- : A light aluminium disc is suspended on a long string in front of the pole of an electromagnet. When an alternating current is passed through the winding of the electromagnet, the disc is repelled STATEMENT- : Due to change of magnetic flux induced (eddy) currents are set up in the disc in the opposite direction. Hence the disc is repelled. 9. STATEMENT- : A capacitor of suitable capacitance can be used in an a.c. circuit in place of the choke coils. STATEMENT- : Average power consume per cycle in an ideal capacitor is zero.. STATEMENT- : A bulb connected in series with a solenoid is lit by a.c. source. If a soft iron core is introduced in the solenoid then the bulb will glow dimmer. STATEMENT- : On introduction of soft iron code in the solenoid, its inductance increases.. STATEMENT- : Using the ordinary ammeter, we can measure the value of alternating current. STATEMENT- : The average value of a.c. in complete cycle is zero. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

18 . STATEMENT- : Using hot wire instrument we can measure the a.c. and d.c. both. STATEMENT- : Both a.c. and d.c. produce heat which is proportional to square of the current. PEMI 8 (Answers) EXCECISE BASED ON NEW PATTEN COMPEHENSION TYPE. a. d 3. b 4. d 5. b 6. d 7. d 8. a 9. d. a. b. a 3. d 4. b 5. c 6. a 7. d 8. b 9. a. a. b. b 3. b 4. b 5. a 6. c 7. a 8. d MATIX-MATCH TYPE. [A-P; B-Q; C-; D-S] MULTIPLE COECT CHOICE TYPE. a, c. b, c, d 3. b, c 4. a, b, d 5. a, b, c, d 6. a, c, d 7. a, d 8. a, b 9. b, c. a, d ASSETION-EASON TYPE. A. A 3. A 4. A 5. A 6. D 7. D 8. A 9. A. A. D. A. A square loop of wire (side a) lies on a table, a distance s from a very long straight wire, which carries a current I, as shown in figure. INITIAL STEP EXECISE (SUBJECTIVE) Find the flux of B through the loop. If someone now pulls the loop directly away from the wire, at speed v, what emf is generated? In what direction (clockwise or anticlockwise) does the current flow?. A square loop of wire with resistance is moved at constant speed v across a region whose sides are twice the length of those of the square loop. Sketch a graph of the external force F needed to move the loop at constant speed, as function of the coordinate x, from x = L to x = +L. Take positive force to be the right. Sketch a graph of the induced current in the loop as a function f(x)? Take anticlockwise current to be positive. 3. The current in an ideal solenoid of radius varies as a function of time. Find the magnitude of induced electric field at points inside, and outside the db solenoid. Express the result in terms of. Also draw the variation of electric field with r. Here r is the distance of a point from the axis of the solenoid. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

19 4. Figure shows a square loop of side L perpendicular to the uniform field of a solenoid Show that at any point on a side the component of the induced electric field along the side is db L A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii a and b. The inner cylinder carries a steady current i, the other cylinder providing the return path for that current. The current sets up a magnetic field between the two cylinders. Calculate the energy stored in the magnetic field for a length l of the cable. 6. In an L C circuit Q max = µc; L = 4 mh; C = µf. Find : (i) the equation for instant charge on the capacitor; (ii) the equation for instant current in the circuit; (iii) Plot the following graphs q versus t, i versus t, U E versus t, U B versus t 7. A pair of parallel horizontal conducting rails of negligible resistance shorted at one end on a table. The distance between the rails is l. A conducting massless rod of resistance can slide on the rails frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the edge of the table. A mass m, tied to the other end of the string, hangs vertically. A constant magnetic field B exists perpendicular to the table. If the system is released from rest, calculate the terminal velocity achieved by the rod, and the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity. 8. A coil A-C-D of radius and number of turns n carries a current i amp, and is placed in the plane of paper. A small conducting loop P of radius r is placed at a distance y from the centre and above the coil A C D. Calculate the induced emf produced in the ring when the ring is allowed to fall freely. Express induced emf in terms of speed of the ring. PEMI 9 9. An inductor of inductance. mh is connected across a charged capacitor of capacitance 5. µf and the resulting L C circuit is set oscillating at its natural frequency. Let q denote the instantaneous charge on the capacitor, and I the current in the circuit. It is found that the maximum value of q is µc. When q = µc what is the value of di/? When q = µc, what is the value of I? Find the maximum value of I When I is equal to one half its maximum value what is the value of q?. Calculate the steady state velocity with which a conducting wire of length l, mass m, and resistance will slide down without friction on two parallel conducting rails of negligible resistance. The bottom ends of the rails and connected as shown in figure. The inclination of the rails to the horizontal is, and a uniform magnetic field of induction B is assumed to exist in the vertical direction.. A square frame with side a and a long straight wire carrying a current I are located in the same plane as shown in the figure. The frame translates to the right with a constant velocity v. Find the emf induced in the frame as function of distance x.. A long straight wire carrying a current I and a U-shaped conductor with sliding connector are located in the same plane as shown in the figure. The connector of length l, and resistance slides to the right with a constant velocity v. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

20 Find the current induced in the loop as a function of separation x between the connector and the straight wire. 3. A 3.56 H inductor is placed in series with a.8 resistor, and an emf of 3.4 V is then suddenly applied across the L combination. At.78 s after the emf is applied what is the rate at which energy is being delivered by the battery? At.78 s, at what rate is energy appearing as thermal energy in the resistor? At.78 s, at what rate is energy being stored in the magnetic field? 4. A long, straight wire has a constant current I. A metal rod of length l moves at velocity v relative to the wire, as shown in figure. What is the potential difference between the ends of the rod 5. A metal disc of radius a = 5 cm rotates with a constant angular velocity = 3 rad/s about its axis. Find the potential difference between the centre and the rim of the disc if the external magnetic field is absent; the external uniform magnetic field of induction B = 5. mt is directed perpendicular to the disc. 6. Find the inductance of a solenoid of length l whose winding is made of copper wire of mass m. The winding resistance is equal to. The solenoid diameter is considerably less than its length. The resistivity and density of copper is and respectively. PEMI 7. Find the inductance of a unit length of a cable consisting of two thin-walled coaxial metallic cylinders if the radius of the outside cylinder is = 3.6 times that of the inside one. The permeability of a medium between the cylinders is assumed to be equal to unity. 8. A superconducting round ring of radius a and inductance L was located in a uniform magnetic field of induction B. The ring plane was parallel to the vector B, and the current in the ring was equal to zero. Then the ring was turned through 9 so that its plane became perpendicular to the field. Find : the current induced in the ring after the turn; the work performed during the turn. 9. Two straight conducting rails form a right angle where their ends are joined. A conducting bar in contact with the rails starts at the vertex at time t = and moves with a constant velocity of 5. m/s along them, as shown in fig. A magnetic field with B =.35 T is directed out of the page. Calculate the flux through the triangle formed by the rails and bar at t = 3. s and the emf around the triangle at that time. If we write the emf as = at n, where a and n are constants, what is the value of n?. A conducting ring of radius a is rotated in a uniform magnetic field B about P in the plane of the paper as shown in the figure. Find the induced emf between P and Q and indicate the polarity of the points P and Q If a resistance is connected between P and Q determine the current through the resistor. Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857

21 . A plane loop as shown in figure is shaped as two squares with sides a = cm and b = cm and is introduced into a uniform magnetic field at right angles to the loop s plane. The magnetic induction varies with time as B = B sin t, where B = mt and = s. PEMI Obtain current through the resistance at resonance in both the circuits. 6. For the circuit shown in fig. current in inductance is.8 A while in capacitance is.6 A. Find the amplitude of the current induced in the loop if its resistance per unit length is equal to = 5 m m.. An inductor 3 henry, a capacitor µf and a resistor 5 are connected in series across a source of emf V = sin 34t. Find the energy dissipated in the circuit in minutes. If resistance is removed from the circuit and the value of inductance is doubled, then find the variation of current with time in the new circuit. 3. A current of 4 A flows in a coil when connected to a V dc source. If the same coil is connected to a V, 5 rad/s ac source a current of.4 A flows in the circuit. Determine the inductance of the coil. Also find the power developed in the circuit if a 5 µf capacitor is connected in series with the coil. 4. A choke coil is needed to operate an arc lamp at 6 V (rms) and 5 Hz. The arc lamp has an effective resistance of 5 when running at A(rms). Calculate the inductance of the choke coil. If the same arc lamp is to be operated on 6 V (dc), what additional resistance is required? Compare the power losses in both cases. 5. An ac source is connected to two circuits as shown in fig. (A) and (B). What is the current drawn from the source? 7. For the circuit shown in figure Find the expressions for the impendence of the circuit and phase of current. 8. A box contains L, C and. When 5 V dc is applied to the terminals of the box, a current of. flows in the circuit. When an ac source of 5 V rms at 5 rad/s is connected, a current of.5 A rms flows. It is observed that the current rises with frequency and becomes maximum at 45 rad/s. Find the values of L, C and. Draw the circuit diagram. 9. An LC circuit has L = mh, = 3 and C = µf connected in series to a source of 5 cos t V. Calculate the current amplitude and the average power dissipates per cycle at a frequency that is % lower than the resonance frequency. 3. A series LC circuit containing a resistance of has angular resonance frequency 4 5 rad s. At resonance the voltages across resistance and inductance are 6 V and 4 V respectively. Find the values of L and C. At what frequency the current in the circuit lags the voltage by 45? Einstein Classes, Unit No., 3, Vardhman ing oad Plaza, Vikas Puri Extn., Outer ing oad New Delhi 8, Ph. : , 857