EXERCISE–IQ.1 The horizontal component of the earth’s magnetic field at a place is 3 × 10 –4 T and the dip is tan–1(4/3). A metal rod of length 0.25 m placed in the north-south position is moved at a constant speed of 10cm/s towards the east. Find the e.m.f. induced in the rod. r V Vx iˆ Vy ˆj Q.2 A wire forming one cycle of sine curve is moved in x-y plane with velocity r B B0 kˆ exist a magnetic field. Find the motional . There emf develop across the ends PQ of wire. Q.3 A conducting circular loop is placed in a uniform magnetic field of 0.02 T, with its plane perpendicular to the field . If the radius of the loop starts shrinking at a constant rate of 1.0 mm/s, then find the emf induced in the loop, at the instant when the radius is 4 cm. Q.4 A rectangular loop with a sliding connector of length l = 1.0 m is situated in a uniform magnetic field B = 2T perpendicular to the plane of loop. Resistance of connector is r = 2. Two resistances of 6 and 3 are connected as shown in figure. Find the external force required to keep the connector moving with a constant velocity v = 2m/s. Q.5 Two concentric and coplanar circular coils have radii a and b(>>a)as shown in figure. Resistance of the inner coil is R. Current in the outer coil is increased from 0 to i, then find the total charge circulating the inner coil. Q.6 A horizontal wire is free to slide on the vertical rails of a conducting frame as shown in figure. The wire has a mass m and length l and the resistance of the circuit is R. If a uniform magnetic field B is directed perpendicular to the frame, then find the terminal speed of the wire as it falls under the force of gravity. Q.7 A metal rod of resistance 20 is fixed along a diameter of a conducting ring of radius 0.1 m and r B kˆ lies on x-y plane. There is a magnetic field = (50T) . The ring rotates with an angular velocity = 20 rad/sec about its axis. An external resistance of 10 is connected across the centre of the ring and rim. Find the current through external resistance. Q.8 A triangular wire frame (each side = 2m) is placed in a region of time variant magnetic field having 3 dB/dt = T/s. The magnetic field is perpendicular to the plane of the triangle. The base of the triangle AB has a resistance 1 while the other two sides have resistance 2 each. The magnitude of potential difference between the points A and B will be Q.9 There exists a uniform cylindrically symmetric magnetic field directed along the axis of a cylinder but varying with time as B = kt. If an electron is released from rest in this field at a distance of ‘r’ from the axis of cylinder, its acceleration, just after it is released would be (e and m are the electronic charge and mass respectively) Q.10 A uniform but time varying magnetic field B = Kt – C ; (0 t C/K), where K and C are constants and t is time, is applied perpendicular to the plane of the circular loop of radius ‘a’ and resistance R. Find the total charge that will pass around the loop. Q.11 A charged ring of mass m = 50 gm, charge 2 coulomb and radius R = 2m is placed on a smooth horizontal surface. A magnetic field varying with time at a rate of (0.2 t) Tesla/sec is applied on to the ring in a direction normal to the surface of ring. Find the angular speed attained in a time t 1 = 10 sec. Assume that the magnetic field is cylindrically symmetric and covering the entire ring. Q.12 In the given current, find the ratio of i 1 to i2 where i1 is the initial (at t = 0) current and i 2 is steady state (at t = ) current through the battery. L RCV Q.13 Find the dimension of the quantity Q.14 In the circuit shown, initially the switch is in position 1 for a long time. Then the switch is shifted to position 2 for a long time. Find the total heat produced in R 2. Q.15 In the circuit shown in figure switch S is closed at time t = 0. Find the charge which passes through the battery in one time constant. Q.16 Two coils, 1 & 2, have a mutual inductance = M and resistances R each. A current flows in coil 1, which varies with time as: I1 = kt2 , where K is a constant and 't' is time. Find the total charge that has flown through coil 2, between t = 0 and t = T. Q.17 In a L–R decay circuit, the initial current at t = 0 is I. Find the total charge that has flown through the resistor till the energy in the inductor has reduced to one–fourth its initial value. Q.18 A capacitor C with a charge Q0 is connected across an inductor through a switch S. If at t = 0, the switch is closed, then find the instantaneous charge q on the upper plate of capacitor. Q.19 An inductor of inductance 2.0mH,is connected across a charged capacitor of capacitance 5.0F,and the resulting LC 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 200 C. when Q=100C,what is the value of ? (a) , where symbols have usual meaining. (b) (c) (d) when Q=200 C ,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.20 Find the value of an inductance which should be connected in series with a capacitor of 5 F, a resistance of 10 and an ac source of 50 Hz so that the power factor of the circuit is unity. Q.21 In an L-R series A.C circuit the potential difference across an inductance and resistance joined in series are respectively 12 V and 16V. Find the total potential difference across the circuit. Q.22 A 50W, 100V lamp is to be connected to an ac mains of 200V, 50Hz. What capacitance is essential to be put in series with the lamp. EXERCISE–II Q.1 Two straight conducting rails form a right angle where their ends are joined. A conducting bar contact with the rails starts at vertex at the time t = 0 & moves symmetrically with a constant velocity of 5.2 m/s to the right as shown in figure. A 0.35 T magnetic field points out of the page. Calculate: (i) (ii) (iii) The flux through the triangle by the rails & bar at t = 3.0 s. The emf around the triangle at that time. In what manner does the emf around the triangle vary with time . Q.2 A wire is bent into 3 circular segments of radius r = 10 cm as shown in figure . Each segment is a quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane & ca lying in the zx plane. z c r r (i) (ii) Q.3 x b y a if a magnetic field B points in the positive x direction, what is the developed in the wire, when B increases at the rate of 3 mT/s ? what is the direction of the current in the segment bc. magnitude of the emf A long straight wire is arranged along the symmetry axis of a toroidal coil of rectangular crosssection, whose dimensions are given in the figure. The number of turns on the coil is N, and relative permeability of the surrounding medium is unity. Find the amplitude of the emf induced in this coil, if the current i = im cos t flows along the straight wire. r B Q.4 A uniform magnetic field fills a cylindrical volumes of radius R. A metal rod CD of length l is placed inside the cylinder along a chord of the circular cross-section as shown in the figure. If the magnitude of magnetic field increases in the direction of field at a constant rate dB/dt, find the magnitude and direction of the EMF induced in the rod. Q.5 A variable magnetic field creates a constant emf E in a conductor ABCDA. The resistances of portion ABC, CDA and AMC are R 1, R2 and R3 respectively. What current will be shown by meter M? The magnetic field is concentrated near the axis of the circular conductor. Q.6 In the circuit shown in the figure the switched S 1 and S2 are closed at time t = 0. After time t = (0.1) ln 2 sec, switch S2 is opened. Find the current in the circuit at time t = (0.2) Q.7 Find the values of i1 and i2 (i) (ii) immediately after the switch S is closed. long time later, with S closed. ln 2 sec. (iii) (iv) immediately after S is open. long time after S is opened. Q.8 A rectangular frame ABCD made of a uniform metal wire has a straight connection between E & F made of the same wire as shown in the figure. AEFD is a square of side 1 m & EB = FC = 0.5 m. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the place of the paper & normal to it . The rate of change of the magnetic field is 1 T/s, the resistance per unit length of the wire is 1 /m. Find the current in segments AE, BE & EF. Q.9 Two parallel vertical metallic rails AB & CD are separated by 1 m. They are connected at the two ends by resistance R1 & R2 as shown in the figure. A horizontally metallic bar L of mass 0.2 kg slides without friction, vertically down the rails under the action of gravity. There is a uniform horizontal magnetic field of 0.6T perpendicular to the plane of the rails, it is observed that when the terminal velocity is attained, the power dissipated in R 1 & R2 are 0.76 W & 1.2 W respectively. Find the terminal velocity of bar L & value R1 & R2. Q.10 A metal rod OA of mass m & length r is kept rotating with a constant angular speed in a vertical plane about a horizontal axis at the end O. The free end A is arranged to slide without friction along a fixed conducting circular ring in the same plane as that of rotation. A uniform & constant r B magnetic induction is applied perpendicular & into the plane of rotation as shown in figure. An inductor L and an external resistance R are connected through a switch S between the point O & a point C on the ring to form an electrical circuit. Neglect the resistance of the ring and the rod. Initially, the switch is open. (a) (b) What is the induced emf across the terminals of the switch? (i) Obtain an expression for the current as a function of time after switch S is closed. (ii) Obtain the time dependence of the torque required to maintain the constant angular speed, given that the rod OA was along the positive X-axis at t = 0. Q.11 A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on a table. The distance between the rails is L. A conducting massless rod of resistance R 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: (i) (ii) the terminal velocity achieved by the rod. the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity. Q.12 In the LR circuit shown, what is the variation of the current I as a function of time? The switch is closed at time t = 0 sec. Q.13 A current of 4 A flows in a coil when connected to a 12 V dc source. If the same coil is connected to a 12V, 50 rad/s ac source a current of 2.4 A flows in the circuit. Determine the inductance of the coil. Also find the power developed in the circuit if a 2500 F capacitor is connected in series with the coil. Q.14 An LCR series circuit with 100 resistance is connected to an ac source of 200 V and angular frequency 300 rad/s. When only the capacitance is removed, the current lags behind the voltage by 60°. When only the inductance is removed, the current leads the voltage by 60°. Calculate the current and the power dissipated in the LCR circuit. Q.15 A series LCR circuit containing a resistance of 120 has angular resonance frequency 4 × 10 5 rad s–1. At resonance the voltages across resistance and inductance are 60 V and 40 V respectively. Find the values of L and C. At what frequency the current in the circuit lags the voltage by 45°? EXERCISE–III Q.1 Two identical circular loops of metal wire are lying on a table without touching each other. Loop -A carries a current which increases with time. In response, the loop-B [JEE ’99] (A) remains stationary (C) is repelled by the loop-A (B) is attracted by the loop-A (D) rotates about its CM, with CM fixed Q.2 A coil of inductance 8.4 mH and resistance 6 is connected to a 12V battery. The current in the coil is 1.0 A at approximately the time (A) 500 s (B) 20 s (C) 35 ms (D) 1 ms [ JEE ’99 ] Q.3 A circular loop of radius R, carrying current I, lies in x-y plane with its centre at origin. The total magnetic flux through x-y plane is (A) directly proportional to I (B) directly proportional to R (C) directly proportional to R2 (D) zero [JEE ’99] Q.4 A magnetic field B = (B 0y / a) is into the plane of paper in the +z direction. B 0 and a are positive constants. A square loop EFGH of side a, mass m and resistance R, in x-y plane, starts falling under the influence of gravity. Note the directions of x and y axes in the figure. Find (a) (b) (c) the induced current in the loop and indicate its direction, the total Lorentz force acting on the loop and indicate its direction, an expression for the speed of the loop, v(t) and its terminal value. Q.5 Two circular coils can be arranged in any of the three situations shown in the figure. Their mutual inductance will be kˆ (A) maximum in situation (a) (B) maximum in situation (b) (C) maximum in situation (c) (D) the same in all situations Q.6 [JEE ’99] [JEE ’2001, (Scr)] An inductor of inductance L = 400 mH and resistors of resistances R 1 = 2 and R2 = 2 are connected to a battery of e.m.f. E = 12V as shown in the figure. The internal resistance of the battery is negligible. The switch S is closed at time t = 0. What is the potential drop across L as a function of time? After the steady state is reached, the switch is opened. What is the direction and the magnitude of current through R1 as a function of time? [JEE ’2001] Q.7 As shown in the figure, P and Q are two coaxial conducting loops separated by some distance. When the switch S is closed, a clockwise current I P flows in P (as seen by E) and an induced current IQ1 flows in Q. The switch remains closed for a long time. When S is opened, a current I Q2 flows in Q. Then the directions of IQ1 adn IQ2 (as seen by E) are: (A) respectively clockwise and anti-clockwise (C) both anti-clockwise (B) both clockwise (D) respectively anti-clockwise and clockwise [JEE 2002(Scr), 3] Q.8 A short -circuited coil is placed in a time varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the radius of cross-section of wire is halved, the electrical power dissipated would be [JEE 2002(Scr), 3] (A) halved (B) the same (C) doubled (D) quadrupled Q.9 A square loop of side 'a' with a capacitor of capacitance C is located between two current carrying long parallel wires as shown. The value of I in the is given as I = I 0sint. (a) (b) calculate maximum current in the square loop. Draw a graph between charge on the lower plate of the capacitor v/s time. Q.10 The variation of induced emf ) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as (A) (B) (C) [JEE 2003] (D) [JEE 2004(Scr)] Q.11 In an LR series circuit, a sinusoidal voltage V = V o sin t is applied. It is given that L = 35 mH, R 2 = 11 , Vrms = 220 V, = 50 Hz and = 22/7. Find the amplitude of current in the steady state and obtain the phase difference between the current and the voltage. Also plot the variation of current for one cycle on the given graph. [JEE 2004] V O T/4 T/2 3T/4 T t Q.12 An infinitely long cylindrical conducting rod is kept along + Z direction. A constant magnetic field is also present in + Z direction. Then current induced will be (A) 0 (B) along +z direction (C) along clockwise as seen from + Z (D) along anticlockwise as seen from + Z [JEE’ 2005 (Scr)] Q. 13 A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d (d <<R) and length L. A variable current i = i 0sin t flows through the coil. If the resistivity of the material of cylindrical shell is , find the induced current in the shell. [JEE 2005 ] Q.14 In the given diagram, a line of force of a particular force field is shown. Out of the following options, it can never represent (A) an electrostatic field (B) a magnetostatic field (C) a gravitational field of a mass at rest (D) an induced electric field 2006] [JEE Comprehension-I The capacitor of capacitance C can be charged (with the help of a resistance R) by a voltage source V, by closing switch S 1 while keeping switch S 2 open. The capacitor can be connected in series with an inductor ‘L’ by closing switch S2 and opening S1. V R S1 C S2 L Q.15 Initially, the capacitor was uncharged. Now, switch S 1 is closed and S2 is kept open. If time constant of this circuit is , then (A) after time interval , charge on the capacitor is CV/2 (B) after time interval 2, charge on the capacitor is CV(1–e–2) (C) the work done by the voltage source will be half of the heat dissipated when the capacitor is fully charged. (D) after time interval 2, charge on the capacitor is CV(1–e–1) [JEE 2006] Q.16 After the capacitor gets fully charged, S1 is opened and S2 is closed so that the inductor is connected in series with the capacitor. Then, (A) at t = 0, energy stored in the circuit is purely in the form of magnetic energy (B) at any time t > 0, current in the circuit is in the same direction (C) at t > 0, there is no exchange of energy between the inductor and capacitor V (D) at any time t > 0, instantaneous current in the circuit may Q.17 C L [JEE 2006] If the total charge stored in the LC circuit is Q0, then for t 0 t LC 2 Q Q0 cos (A) the charge on the capacitor is t LC 2 Q Q0 cos (B) the charge on the capacitor is (C) the charge on the capacitor is d 2Q Q LC 2 dt 1 d 2Q Q 2 LC dt (D) the charge on the capacitor is [JEE 2006] Q.18 Match the following Columns Column 1 Column 2 (A) Dielectric ring uniformly charged (B) Dielectric ring uniformly charged rotating with angular velocity . (C) Constant current i0 in ring (D) Current i = i0 cos t in ring (P) Time independent electrostatic field out of system (Q) Magnetic field (R) Induced electric field (S) Magnetic moment [JEE 2006] Comprehension –II Magler Train: This train is based on the Lenz law and phenomena of electromagnetic induction. In this there is a coil on a railway track and magnet on the base of train. So as train is deviated then as is move down coil on track repel it and as it move up then coil attract it. Disadvantage of magler train is that as it slow down the forces decreases and as it moves forward so due to Lenz law coil attract it backward. Due to motion of train current induces in the coil of track which levitate it. Q.19 What is the advantage of the train? (A) Electrostatic force draws the train (C) Electromagnetic force draws the train Q.20 What is the disadvantage of the train? (A) Train experience upward force due to Lenz's law. (B) Friction force create a drag on the train. (C) Retardation (D) By Lenz's law train experience a drag 2006] Q.21 Q.22 Which force causes the train to elevate up (A) Electrostatic force (C) magnetic force 2006] [JEE 2006] (B) Gravitational force is zero. (D) Dissipative force due to friction are absent (B) Time varying electric field (D) Induced electric field [JEE [JEE STATEMENT-1 A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. The rod goes through a conducting ring as shown in the figure. The ring can float at a certain height above the coil because STATEMENT-2 In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the upward direction. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True [JEE 2007] Q.23 The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. I 1 and I2 are the currents in the segments ab and cd. Then, [JEE-2009] × c× × × × × × × × b× × × × × × × × × × × × × × × × × × × × × a d (A) I1 > I2 (B) I1 < I2 (C) I1 is in the direction ba and I2 is in the direction cd (D) I1 is in the direction ab and I2 is in the direction dc Q.24 Two metallic rings A and B, identical in shape and size but having different resistivities A and B, are kept on top of two identical solenoids as shown in the figure. When current I is switched on in both the solenoids in identical manner, the rings A and B jump to heights h A and hB, respectively, with hA > hB. The possible relation(s) between their resistivities and their masses m A and mB is(are) [JEE-2009] A B (A) A > B and mA = mB (B) A < B and mA = mB (C) A > B and mA > mB (D) A < B and mA < mB OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. Q.1 An electron is moving in a circular orbit of radius R with an angular acceleration . At the centre of the orbit is kept a conducting loop of radius r, (r <<R). The e.m.f induced in the smaller loop due to the motion of the electron is (A) zero, since charge on electron in constant (B) 0 er 2 4R (C) Q.2 Q.3 0 er 2 4 R (D) none of these a conducting loop of radius R is present in a uniform magnetic field B perpendicular the plane of the ring. If radius R varies as a function of time ‘t’, as R = R 0+ t. The e.m.f induced in the loop is (A) 2(R0 + t)B clockwise (B) (R0 + t)B clockwise (C) 2(R0 + t)B anticlockwise (D) zero A square wire loop of 10.0 cm side lies at right angles to a uniform magnetic field of 20T. A 10 V light bulb is in a series with the loop as shown in the fig. The magnetic field is decreasing steadily to zero over a time interval t. The bulb will shine with full brightness if t is equal to (A) 20 ms (C) 2 ms Q.4 (B) 0.02 ms (D) 0.2 ms A rectangular loop with a sliding connector of length 10 cm is situated in uniform magnetic field perpendicular to plane of loop. The magnetic induction is 0.1 tesla and resistance of connector (R) is 1 ohm. The sides AB and CD have resistances 2 ohm and 3 ohm respectively. Find the current in the connector during its motion with constant velocity one metre/sec. (A) 1 110 A (B) 1 220 A (C) 1 55 A (D) 1 440 A Q.5 The dimensions of permeability of free space can be given by (A) [MLT–2 A–2] (B) [MLA–2] (C) [ML–3 T2 A2](D) [MLA–1] Q.6 A closed planar wire loop of area A and arbitrary shape is placed in a uniform magnetic field of magnitude B, with its plane perpendicular to magnetic field. The resistance of the wire loop is R. The loop is now turned upside down by 180° so that its plane again becomes perpendicular to the magnetic field. The total charge that must have flowed through the wire ring in the process is (A) < AB/R (B) = AB/R (C) = 2AB/R (D) None Q.7 A vertical bar magnet is dropped from position on the axis of a fixed metallic coil as shown in fig I. In fig - II the magnet is fixed and horizontal coil is dropped. The acceleration of the magnet and coil are a1 and a2 respectively then (A) a1 > g , a2 > g (C) a1 < g , a2 < g (B) a1 > g , a2 < g (D) a1 < g , a2 > g fig - I fig-II Q.8 Two identical coaxial circular loops carry a current i each circulating in the same direction. If the loops approach each other (A) the current in each will decrease (B) the current in each will increase (C) the current in each will remain the same (D) the current in one will increase and in other will decrease Q.9 In the arrangement shown in given figure current from A to B is increasing in magnitude. Induced current in the loop will (A) have clockwise direction (B) have anticlockwise direction (C) be zero (D) oscillate between clockwise and anticlockwise Q.10 An electric current i1 can flow either direction through loop (1) and induced current i 2 in loop (2). Positive i1 is when current is from 'a' to 'b' in loop (1) and positive i 2 is when the current is from 'c' to 'd' in loop (2) In an experiment, the graph of i2 against time 't' is as shown below Which one(s) of the following graphs could have caused i2 to behave as give above. (A) (B) (C) (D) (E) Q.11 Q.12 The figure shows an isosceles triangle wire frame with apex angle equal to /2. The frame starts entering into the region of uniform magnetic field B with constant velocity v at t= 0. The longest side of the frame is perpendicular to the direction of velocity. If i is the instantaneous current through the frame then choose the alternative showing the correct variation of i with time. (A) (B) (C) (D) A thin wire of length 2m is perpendicular to the xy plane. It is moved with velocity r v (2iˆ 3 ˆj kˆ) m / s through a region of magnetic induction potential difference induced between the ends of the wire: (A) 2 volts (B) 4 volts (C) 0 volts r B (iˆ 2 ˆj ) Wb / m 2 . Then (D) none of these Q.13 A long metal bar of 30 cm length is aligned along a north south line and moves eastward at a speed of 10 ms–1. A uniform magnetic field of 4.0 T points vertically downwards. If the south end of the bar has a potential of 0 V, the induced potential at the north end of the bar is (A) + 12 V (B) – 12 V (C) 0 V (D) cannot be determined since there is not closed circuit Q.14 A conducting rod moves with constant velocity perpendicular to the long, straight wire carrying a current I as shown compute that the emf generated between the ends of the rod. (A) Q.15 (B) Bl 2 vx R (B) (A) (D) 0 Il 4 r 2B 2 l 2vx R (C) 4B 2 l 2 vx R (D) none ( F1 F2 ) mL eM (B) mL eM (C) [ ] ln [F1/F2] (D) None Two parallel long straight conductors lie on a smooth surface. Two other parallel conductors rest on them at right angles so as to form a square of side a initially. A uniform magnetic field B exists at right angles to the plane containing the conductors. They all start moving out with a constant velocity v. If r is the resistance per unit length of the wire the current in the circuit will be (A) Q.18 (C) 20 Il r A metallic rod of length L and mass M is moving under the action of two unequal forces F 1 and F2 (directed opposite to each other) acting at its ends along its length. Ignore gravity and any external magnetic field. If specific charge of electrons is (e/m), then the potential difference between the ends of the rod is steady state must be F1 F2 mL eM Q.17 0 Il 2 r A square loop of side a and resistance R is moved in the region of uniform magnetic field B(loop remaining completely inside field) ,with a velocity v through a distance x . The work done is : (A) Q.16 0 Il r Bv r (B) Br v (C) Bvr (D) Bv There is a uniform magnetic field B normal to the xy plane. A conductor ABC has length AB = l1, parallel to the x-axis, and length BC = l2, parallel to the y-axis. ABC moves in the xy plane with vx iˆ v y ˆj velocity (A) vxl1 + vyl2 (C) vxl2 – vyl1 . The potential difference between A and C is proportional to (B) vxl2 + vyl1 (D) vxl1 – vyl2 Q.19 A conducting rod PQ of length 5 m oriented as shown in figure is moving with velocity (2 m/s) (3 ˆj 4kˆ) without any rotation in a uniform magnetic field (A) 32 Volts Q.20 iˆ (B) 40 Volt Tesla. Emf induced in the rod is (C) 50 Volt (D) none x ˆ 1 k a r B The magnetic field in a region is given by = B0 . A square loop of edge length d is placed with its edge along x & y axis. The loop is moved with constant r V V0iˆ velocity (A) Q.21 . The emf induced in the loop is V0 B0 d a 2 (B) (C) V0 B0 a 2 d (D) None When a ‘J’ shaped conducting rod is rotating in its own plane with constant angular velocity w, about one of its end P, in a uniform magnetic field directed normally into the plane of paper) then magnitude of emf induced across it will be L l 2 (A) B Q.22 V0 B0 d 2 2a 2 (B) 1 B L2 2 (C) 1 B ( L2 l2 ) 2 (D) 1 B l2 2 A metal disc rotates freely, between the poles of a magnet in the direction indicated. Brushes P and Q make contact with the edge of the disc and the metal axle. What current, if any, flows through R? (A) a current from P to Q (B) a current from Q to P (C) no current, because the emf in the disc is opposed by the back emf (D) no current, because the emf induced in one side of the disc is opposed by the emf induced in the other side. (E) no current, because no radial emf is induced in the disc Q.23 A ring of resistance 10, radius 10cm and 100 turns is rotated at a rate 100 revolutions per second about its diameter is perpendicular to a uniform magnetic field of induction 10mT. The amplitude of the current in the loop will be nearly (Take : 2 = 10) (A) 200A (B) 2A (C) 0.002A (D) none of these Q.24 A copper rod AB of length L, pivoted at one end A, rotates at constant angular velocity , at right angles to a uniform magnetic field of induction B. The e.m.f developed between the mid point C of the rod and end B is (A) Q.25 B l2 4 (B) B l2 2 (C) 3Bl 2 4 (D) 3B l 2 8 Figure shows a uniform magnetic field B confined to a cylindrical volume and is increasing at a constant rate. The instantaneous acceleration experienced by an electron placed at P is (A) zero (B) towards right (C) towards left (D) upwards Q.26 For L-R circuit, the time constant is equal to (A) twice the ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance (B) ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance (C) half the ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance (D) square of the ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance Q.27 In the adjoining circuit, initially the switch S is open. The switch ‘S’ is closed at t = 0. The difference between the maximum and minimum current that can flow in the circuit is (A) 2 Amp (B) 3 Amp (C) 1 Amp (D) nothing can be concluded Q.28 The ratio of time constant in build up and decay in the circuit shown in figure is (A) 1 : 1 Q.29 (B) 3 : 2 (C) 2 : 3 (D) 1 : 3 In an L-R circuit connected to a battery of constant e.m.f. E switch S is closed at time t = 0. If e denotes the magnitude of induced e.m.f. across inductor and i the current in the circuite at any time t. Then which of the following graphs shows the variation of e with i ? (A) (B) (C) (D) Q.30 A curren of 2A is increasing at a rate of 4 A/s through a coil of inductance 2H. The energy stored in the inductor per unit time is (A) 2 J/s (B) 1 J/s (C) 16 J/s (D) 4 J/s Q.31 Two identical inductance carry currents that vary with time according to linear laws (as shown in figure). In which of two inductance is the self induction emf greater? (A) 1 (B) 2 (C) same (D) data are insufficient to decide Q.32 The current in the given circuit is increasing with a rate a = 4 amp/s. The charge on the capacitor at an instant when the current in the circuit is 2 amp will be: (A) 4C (C) 6C (B) 5C (D) none of these Q.33 L, C and R represent physical quantities inductance, capacitance and resistance. The combination which has the dimensions of frequency is (A) Q.34 1 and RC 1 R and RC L LC (B) (C) (D) C L A coil of inductance 5H is joined to a cell of emf 6V through a resistance 10 at time t = 0. The emf across the coil at time t = ln (A) 3V (B) 1.5 V Q.35 R L 2 s is: (C) 0.75 V (D) 4.5 V The network shown in the figure is part of a complete circuit. If at a certain instant, the current I is 5A and it is decreasing at a rate of 103 As–1 then VB–VA equals (A) 20 V (B) 15 V (C) 10 V (D) 5 V Q.36 In Problem 35, if I is reversed in direction, then VB – VA equals (A) 5 V (B) 10 V (C) 15 V (D) 20 V Q.37 A small coil of radius r is placed at the centre of a large coil of radius R, where R >> r. The coils are coplanar. The coefficient of mutual inductance between the coils is (A) Q.38 0 r 2R (B) 0 r 2 2R (D) 0 r 2R2 In the circuit shown, the cell is ideal. The coil has an inductance of 4H and zero resistance. F is a fuse of zero resistance and will blow when the current through it reaches 5A. The switch is closed at t = 0. The fuse will blow: (A) just after t=0 (C) after 5s Q.39 (C) 0 r 2 2R 2 (B) after 2s (D) after 10s In the circuit shown, X is joined to Y for a long time, and then X is joined to Z. The total heat produced in R2 is: (A) LE 2 2 R12 (B) LE 2 2 R22 (C) LE 2 2 R1R2 (D) LE 2 R2 2 R12 Q.40 The circuit shown has been operating for a long time. The instant after the switch in the circuit labeled S is opened, what is the voltage across the inductor V L and which labeled point (A or B) of the inductor is at a higher potential ? Take R1 = 4.0 , R2 = 8.0 , and L = 2.5 H. L A S B = 12v R R 1 2 (A) VL = 12 V ; Point A is at the higher potential (B) VL = 12 V ; Point B is at the higher potential (C) VL = 6V ; Point A is at the higher potential (D) VL = 6V ; Point B is at the higher potential Q.41 An induction coil stores 32 joules of magnetic energy and dissipates energy as heat at the rate of 320 watts when a current of 4 amperes is passed through it. Find the time constant of the circuit when the coil is joined across a battery. (A) 0.2 s (B) 0.1 s (C) 0.3 s (D) 0.4 s Q.42 The figure shows a part of a complete circuit. The potential difference V B – VA when the current I is 5A and is decreasing at a rate of 103 As–1 is given by (A) 15 V (B) 10 V (C) –15 V (D) 20 V Paragraph for question nos. 43 to 45 In a series L-R circuit, connected with a sinusoidal ac source, the maximum potential difference across L and R are respectively 3 volts and 4 volts. Q.43 At an instant the potential difference across resistor is 2 volts. The potential difference in volt, across the inductor at the same instant will be : (A) 3 cos 30° (B) 3 cos 60° (C) 3 cos 45° (D) None of these Q.44 At the same instant, the magnitude of the potential difference in volt, across the ac source may be (A) 43 3 (B) 43 3 2 1 (C) 3 2 2 (D) 3 2 Q.45 If the current at this instant is decreasing the magnitude of potential difference at that instant across the ac source is (A) Increasing (B) Decreasing (C) Constant (D) Cannot be said Q.46 An inductor coil stores U energy when i current is passed through it and dissipates energy at the rate of P. The time constant of the circuit, when this coil is connected across a battery of zero internal resistance is (A) Q.47 (B) U P (C) 0 R 2 (B) 0 R 2 The power factor of the circuit is 1/ (C) 2 (D) 2P U 0 2 (D) 0 . The capacitance of the circuit is equal to (A) 400 F (C) 500 F Q.49 2U P A long straight wire is placed along the axis of a circular ring of radius R. The mutual inductance of this system is (A) Q.48 4U P (B) 300 F (D) 200 F In the circuit, as shown in the figure, if the value of R.M.S current is 2.2 ampere, the power factor of the box is 1 2 (A) (B) 1 (C) 3 2 (D) 1 2 Q.50 When 100 V DC is applied across a solenoid a current of 1 A flows in it. When 100 V AC is applied across the same coil, the current drops to 0.5 A. If the frequency of the AC source is 50 Hz, the impedance and inductance of the solenoid are: (A) 100, 0.93 H (B) 200, 1.0 H (C) 10, 0.86H (D) 200, 0.55 H Q.51 In ac circuit when ac ammeter is connected it reads i current if a student uses dc ammeter in place of ac ammeter the reading in the dc ammeter will be: i 2 (A) (B) 2 i (C) 0.637 i (D) zero Q.52 The phase difference between current and voltage in an AC circuit is /4 radian. If the frequency of AC is 50 Hz, then the phase difference is equivalent to the time difference : (A) 0.78 s (B) 15.7 ms (C) 0.25 s (D) 2.5 ms Q.53 Power factor of an L-R series circuit is 0.6 and that of a C–R series circuit is 0.5. If the element (L, C, and R) of the two circuits are joined in series the power factor of this circuit is found to be 1. The ratio of the resistance in the L-R circuit to the resistance in the C–R circuit is 4 3 3 (A) 6/5 Q.54 2 (D) 2 2 3 A (B) (C) 4 (D) None If I1, I2, I3 and I4 are the respective r.m.s. values of the time varying currents as shown in the four cases I, II, III and IV. Then identify the correct relations. (A) I1 = I2 = I3 = I4 Q.56 (C) The effective value of current i = 2 sin 100 t + 2 sin(100 t + 30°) is : (A) Q.55 (B) 5/6 (B) I3 > I1 = I2 > I4 (C) I3 > I4 > I2 = I1 (D) I3 > I2 > I1 > I4 In series LR circuit XL = 3R. Now a capacitor with XC = R is added in series. Ratio of new to old power factor is 1 2 (A) 1 Q.57 (B) 2 (C) (D) 2 The current I, potential difference VL across the inductor and potential difference V C across the capacitor in circuit as shown in the figure are best represented vectorially as (A) Q.58 3 3 4 (B) (C) (D) In the shown AC circuit phase different between currents I 1 and I2 is (A) 2 –tan–1 xL R (B) tan–1 xL xC R (C) 2 + tan–1 xL R (D) tan–1 xL xC R 2 + Q.59 In a series R-L-C circuit, the frequency of the source is half of the resonance frequency. The nature of the circuit will be (A) capacitive (B) inductive (C) purely resistive (D) data insufficient ASSERTION AND REASON Q.1 Statement-1: When a circuit having large inductance is switched off sparking occurs at the switch. di dt Statement-2: Emf induced in an inductor is given by || = L. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. Q.2 Statement-1: Statement-2: When resistance of rheostat is increased, clockwise current is induced in the ring. Magnetic flux through the ring is out of the page and decreasing. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. Q.3 Statement-1: Peak voltage across the resistance can be greater than the peak voltage of the source in an series LCR circuit. Statement-2: Peak voltage across the inductor can be greater than the peak voltage of the source in an series LCR circuit. (A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement1. (B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1. (C) Statement-1 is true, statement-2 is false. (D) Statement-1 is false, statement-2 is true. ONE OR MORE THAN ONE OPTION MAY BE CORRECT Take approx. 3 minutes for answering each question. Q.1 The dimension of the ratio of magnetic flux and the resistance is equal to that of : (A) induced emf (B) charge (C) inductance (D) current Question No. 2 to 5 (4 questions) The adjoining figure shows two different arrangements in which two square wire frames of same resistance are placed in a uniform constantly decreasing magnetic field B. Q.2 The value of magnetic flux in each case is given by (A) Case I: = (L2 + 2)B; Case II: = (L2 – 2)B (B) Case I: = (L2 + 2)B; Case II: = (L2 + 2)B (C) Case I: = (L2 + 2)B; Case II: = (L2 – 2)B (D) Case I: = (L + )2B; Case II: = (L – )2B Q.3 The direction of induced current in the case I is (A) from a to b and from c to d (B) from a to b and from f to e (C) from b to a and from d to c (D) from b to a and from e to f Q.4 The direction of induced current in the case II is (A) from a to b and from c to d (B) from b to a and from f to e (C) from b to a and from c to d (D) from a to b and from d to c Q.5 If I1 and I2 are the magnitudes of induced current in the cases I and II, respectively, then (A) I1 = I2 (B) I1 > I2 (C) I1 < I2 (D) nothing can be said Q.6 Figure shown plane figure made of a conductor located in a magnetic field along the inward normal to the plane of the figure. The magnetic field starts diminishing. Then the induced current (A) at point P is clockwise (C) at point Q is clockwise Q.7 (B) at point Q is anticlockwise (D) at point R is zero A conducting wire frame is placed in a magnetic field which is directed into the paper. The magnetic field is increasing at a constant rate. The directions of induced currents in wires AB and CD are (A) B to A and D to C (C) A to B and D to C (B) A to B and C to D (D) B to A and C to D Q.8 A bar magnet is moved along the axis of copper ring placed far away from the magnet. Looking from the side of the magnet, an anticlockwise current is found to be induced in the ring. Which of the following may be true? (A) The south pole faces the ring and the magnet moves towards it. (B) The north pole faces the ring and the magnet moves towards it. (C) The south pole faces the ring and the magnet moves away from it. (D) The north pole faces the ring and the magnet moves away from it. Q.9 Two circular coils P & Q are fixed coaxially & carry currents I 1 and I2 respectively (A) if I2 = 0 & P moves towards Q, a current in the same direction as I1 is induced in Q (B) if I1 = 0 & Q moves towards P, a current in the opposite direction to that of I2 isinduced in P. (C) when I1 0 and I2 0 are in the same direction then the two coils tend to move apart . (D) when I1 0 and I2 0 are in opposite directions then the coils tends to move apart. Q.10 AB and CD are smooth parallel rails, separated by a distance l, and inclined to the horizontal at an angle . A uniform magnetic field of magnitude B, directed vertically upwards, exists in the region. EF is a conductor of mass m, carrying a current i. For EF to be in equilibrium, B C l A F D B E (A) i must flow from E to F (C) Bil = mg sin (B) Bil = mg tan (D) Bil = mg Q.11 In the previous question, if B is normal to the plane of the rails (A) Bil = mg tan (B) Bil = mg sin (C) Bil = mg cos (D) equilibrium cannot be reached Q.12 A conducting rod PQ of length L = 1.0 m is moving with a uniform speed v = 20 m/s in a uniform magnetic field B = 4.0 T directed into the paper. A capacitor of capacity C = 10 F is connected as shown in figure. Then (A) qA = + 800C and qB = – 800C (B) qA = – 800C and qB = + 800C (C) qA = 0 = qB (D) charged stored in the capacitor increases exponentially with time Q.13 A semicircle conducting ring of radius R is placed in the xy plane, as shown in the figure. A uniform magnetic field is set up along the x–axis. No net emf, will be induced in the ring. if (A) it moves along the x–axis (C) it moves along the z-axis (B) it moves along the y–axis (D) it remains stationary Question No. 14 to 16 (3 questions) A conducting ring of radius a is rotated about a point O on its periphery as shown in the figure in a plane perpendicular to uniform magnetic field B which exists everywhere. The rotational velocity is . Q.14 Choose the correct statement(s) related to the potential of the points P, Q and R (A) VP – VO > 0 and VR – VO < 0 (B) VP = VR > VO (D) VQ – VP = VP – VO (C) VO > VP = VQ Q.15 Choose the correct statement(s) related to the magnitude of potential differences (A) V – V = P O 1 2 Ba2 (C) VQ – VO = 2Ba2 (B) VP – VQ = 1 2 Ba2 (D) VP – VR = 2Ba2 Q.16 Choose the correct statement(s) related to the induced current in the ring (A) Current flows from Q P R Q (B) Current flows from Q R Q (C) Current flows from Q P and from Q R O (D) No current flows Q.17 Current growth in two L-R circuits (b) and (c) as shown in figure (a). Let L 1, L2, R1 and R2 be the corresponding values in two circuits. Then (A) R1 > R2 (B) R1 = R2 (C) L1 > L2 (D) L1 < L2 i (b ) (c) t (a) Q.18 (b) A circuit element is placed in a closed box. At time t=0, constant current generator supplying a current of 1 amp, is connected across the box. Potential difference across the box varies according to graph shown in figure. The element in the box is: (A) resistance of 2 (C) inductance of 2H Q.19 (c) (B) battery of emf 6V (D) capacitance of 0.5F The symbols L, C, R represent inductance, capacitance and resistance respectively. Dimension of frequency are given by the combination 1 LC (A) 1 / RC (B) R / L (C) (D) C / L Q.20 An LR circuit with a battery is connected at t = 0. Which of the following quantities is not zero just after the circuit (A) current in the circuit (B) magnetic field energy in the inductor (C) power delivered by the battery (D) emf induced in the inductor Q.21 For L – R circuit, the time constant is equal to (A) twice the ratio of the energy stored in the magnetic field to the rate of the dissipation of energy in the resistance. (B) the ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance. (C) half of the ratio of the energy stored in the magnetic field to the rate of dissipation of energy in the resistance. (D) square of the ratio of the energy stored in the magnetic field to the rate of dissipation energy in the resistance. Q.22 An inductor L, a resistance R and two identical bulbs B 1 and B 2 are connected to a battery through a switch S as shown in the figure. The resistance of coil having inductance L is also R. Which of the following statement gives the correct description of the happenings when the switch S is closed? (A) The bulb B 2 lights up earlier than B 1 and finally both the bulbs shine equally bright. (B) B 1 light up earlier and finally both the bulbs acquire equal brightness. (C) B 2 lights up earlier and finally B 1 shines brighter than B 2 . (D) B 1 and B 2 light up together with equal brightness all the time. Q.23 Two coil A and B have coefficient of mutual inductance M = 2H. The magnetic flux passing through coil A changes by 4 Weber in 10 seconds due to the change in current in B. Then (A) change in current in B in this time interval is 0.5 A (B) the change in current in B in this time interval is 2A (C) the change in current in B in this time interval is 8A (D) a change in current of 1A in coil A will produce a change in flux passing through B by 4 Weber. Q.24 Two different coils have self inductance 8mH and 2mH. 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 1, V1 and W1 respectively. Corresponding values for the second coil at the same instant are I 2, v2 and W2 respectively. Then: (A) I1 1 I2 4 (B) W2 4 W1 I1 4 I2 V2 1 V1 4 (C) (D) ANSWAR KEY EXERCISE–I Q.1 Q.6 Q.9 10 V Q.2 mgR B 2l2 erk 2m Q.7 VyB0 1 3 A Q.3 5.0 V Q.8 0.4 V Q.4 Q.5 directed along tangent to the circle of radius r, whose centre lies on the axis of cylinder. C a2 R Q.10 2N 0ia 2 2 Rb Q.11 200 rad/sec Q.12 0.8 Q.13 I–1 Q.14 LE 2 2 R12 Q.15 EL eR 2 Q.16 L I 2R kMT2/(R) 1 t LC 2 Q.17 Q.18 q = Q0sin 3 Q.19 Q.21 (a)104A/s (b) 0 (c) 2A (d) 100 C 20 V Q.22 C = 9.2 F Q.20 EXERCISE–II Q.1 (i) 85.22 Tm2; (ii) 56.8 V; (iii) linearly 0 h im N 2 Q.3 ln b a Q.2 Q.4 (i) 2.4 × 105 V (ii) from c to b l dB l2 2 R 2 dt 4 E R1 R1R2 R2 R3 R3 R1 Q.5 Q.7 Q.8 Q.10 Q.6 67/32 A (i) i1 = i2 = 10/3 A, (ii) i1 = 50/11 A ; i2 = 30/11 A, (iii) i1 = 0, i2 = 20/11 A, (iv) i1 = i2 = 0 IEA= 7 22 (a) E = A ; IBE= 1 2 3 11 A ; IFE = Q.13 A Q.9 B r 2 1 e Rt / L 2R Br2 (b) (i) I = mg R B2 Z 2 Q.11 1 22 (i) Vterminal = ; (ii) 0.08 H,17.28 W , (ii) = g 2 Q.12 Q.14 2A, 400W V = 1 ms1, R1 = 0.47 , R2 = 0.30 mgr 2 – cos t + B2 r 4 4R (1 eRt/L) V RtL e R Q.15 0.2 mH, 1 32 F, 8 × 105 rad/s EXERCISE–III Q.1 Q.4 Q.5 Q.9 C Q.2 D Q.3 D (a) i = in anticlockwise direction, v = velocity at time t, (b) Fnett=B02a2V/R, (c) V = A B0 av mgR 1 e B02 a 2 R (a) Imax = Q.6 B02 a 2 t mR 12e–5t, 6e–10t 0 a CI 0 2 ln 2 , (b) Q.7 D Q.8 B Q.10 B Q.11 20 4 A, , V, I Steady state current i = 20sin 1 100t 4 v = 220 2 sin t i = 20 sin (t-/4) 20 O -10 2 Q.12 T T/8 T/4 A 9T/8 T/2 5T/8 Q.13 I= Q.17 Q.20 C D t ( 0 ni0 cos t ) a 2 ( Ld ) 2 R Q.14 A,C Q.15 B Q.16 Q.19 D D Q.24 B,D Q.18 Q.21 (A) P; (B) P, Q, S; (C) Q,S ; (D) Q, R, S C Q.22 A Q.23 D OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Q.1 Q.6 Q.11 Q.16 Q.21 Q.26 Q.31 Q.36 Q.41 Q.46 Q.51 Q.56 B C D A C A A C A C D D Q.2 Q.7 Q.12 Q.17 Q.22 Q.27 Q.32 Q.37 Q.42 Q.47 Q.52 Q.57 C C A A A C C B C D D D Q.1 A Q.2 C Q.1 Q.7 Q.13 Q.18 Q.21 B Q.2 A Q.8 A,B,C,D D A Q.3 Q.8 Q.13 Q.18 Q.23 Q.28 Q.33 Q.38 Q.43 Q.48 Q.53 Q.58 A A A C B B A D A C D C Q.4 Q.9 Q.14 Q.19 Q.24 Q.29 Q.34 Q.39 Q.44 Q.49 Q.54 Q.59 B A B A D A A A B A D A Q.5 Q.10 Q.15 Q.20 Q.25 Q.30 Q.35 Q.40 Q.45 Q.50 Q.55 ASSERTION AND REASON Q.3 D ONE OR MORE THAN ONE OPTION MAY BE CORRECT C Q.3 C Q.4 B Q.5 B Q.6 A, C, D B,C Q.9 B,D Q.10 A,B Q.11 B Q.12 A Q.14 B,D Q.15 C Q.16 D Q.17 B,D Q.19 A,B,C Q.20 D Q.22 A Q.23 B Q.24 A,C,D A D D A B C B D A D B