Introduction

Can Magnetic Field induce an Electric Current ?

Magnetic Flux

What is Magnetic Flux ? Is it exactly similar to Electric Flux ? Because there are no Magnetic Monopoles, so does Gauss Law apply to Magnetic Flux as well ?

Q1

Which of the following field line patterns could represent a magnetic field ?

Q2

Consider a hemispherical closed surface as shown in figure. If the hemisphere is in a uniform magnetic field that makes an angle q with the vertical, calculate the magnetic flux. a) through the flat surface S1 b) through the curved hemispherical surface S2

Faraday\'\'s Law

How is the Induced EMF related to the changing magnetic flux ?

Q3

A stationery coil is placed in a non-uniform magnetic field. Will there be an EMF induced in the coil ?

Lenz\'\'s Law

Can we find the direction of Induced EMF and Induced Current in a coil ? Does conservation of energy give us hint of Lenz?s law ?

Q4

A bar magnet is moving in front of a conducting loop as shown in figures. The line joining the center of the bar and the center of the coil (central axis) is perpendicular to the plane of the coil. What is the direction of Induced EMF in each case ?

Q5

A bar magnet is moved near two parallel circular loops A and B as shown in figures. Then for each case, a) the current in loops will be in the same direction b) the current in loops will be in the opposite direction c) the loops will repel each other d) the loops will attract each other

Q6

Predict the polarity of the capacitor C as shown in figure when S and N poles of two identical magnets approach the coil from opposite sides with equal velocity. The plane of the loop containing the capacitor is perpendicular to the plane of the paper. What happens to the charge on capacitor if the magnets stop moving ?

Q7

The variation of induced EMF with time in a coil, if a short bar magnet is moved along its axis with a constant velocity, is best represented as

Q8

A circular loop of wire is positioned half in and half out of a square region of uniform magnetic field as shown in figure. To induce a clockwise current in loop, it should be moved towards a) left b) right

Q9

A and B are two metallic ring placed at opposite sides of an infinitely long straight conducting wire as shown in figure. If current in the wire is slowly increased, direction of the induced current will be a) clockwise in A anticlockwise in B b) anticlockwise in A clockwise in B c) clockwise in both A and B d) anticlockwise in both A and B If current in the wire is slowly decreased, direction of the induced current will be a) clockwise in A anticlockwise in B b) anticlockwise in A clockwise in B c) clockwise in both A and B d) anticlockwise in both A and B

Q10

A circular coil with area A and resistance R, is placed in uniform magnetic field of magnitude B. The plane of the coil is initially perpendicular to B. The coil is rotated by an angle q about a diameter and charge Q flows through it. Find the relation between the charge Q and change in flux through coil. Repeat the above problem if the plane of coil is initially parallel to Magnetic Field.

Q11

4 conducting rods of length a are hinged as shown in figure. The plane of loop is perpendicular to a magnetic field B. The total resistance of the loop is R. If the loop is suddenly collapsed by horizontal forces as shown, what is the total charge passing through the loop ?

Induced E - 1

How can Magnetic Field induce an Electric Current ? Is there any other type of electric field ?

Induced E - 2

How is the Induced Electric Field different from Electrostatic Electric Field ?

Q12

A uniform but time varying magnetic field B(t) exists in a cylindrical region of radius a and is directed into the plane as shown. Field is increasing at a constant rate dB/dt. Find the magnitude of the induced electric field as a function of distance r from the center of region of magnetic field.

Q13

Figure show two regions of magnetic field of areas A1 and A2, and in each region, magnitude of magnetic field decreases at a constant rate dB/dt. If E is the induced electric field, then find the value of the line integral E. ds over the given paths.

Q14

What will be the direction of induced electric field at points A, B and C What will be the induced EMF between points a) AC b) AB

Q15

Figures show a cylindrical region of radius r where a downward magnetic field B exists, and is increasing at the rate of dB/dt. A conducting rod is placed in different citation as shown. Find the induced emf in rod in each case.

Induced EMF & Potential

Can we define ?Potential Difference? for Induced Electric FIelds? Does it violate the conservation of energy ?

Q16

Two rectangular frames, made of a uniform metal wire, have a straight connection made of the same wire, as shown in figures. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the plane of the screen and normal to it. For the first circuit, rate of change of the magnetic field is b T/s. The resistance per unit length of the wire is R1 W/m. For the second circuit, rate of change of the magnetic field is 1 T/s. The resistance per unit length of the wire is 1 W/m. Find the magnitude and direction of the current in the left, center and right segments for each circuit.

Q17

There is a uniform magnetic field B in a circular region of radius R as shown in figure, whose magnitude changes at the rate of dB/dt. Find the e.m.f induced across the ends of a semi-circular concentric conducting ring of radius a as shown. Can we define a potential difference between the ends of ring ?

Q18

There is a downward magnetic field in a cylindrical region as shown in figure, whose magnitude increases with time. This induces an emf in a conducting wire loop which lights two identical bulbs connected in series along the wire. Now two points diametrically opposed on the wire loop (marked red) are shorted with another wire lying to the right of bulb B in the plane of the screen. After the shorting wire is inserted, a) Bulb A goes out and bulb B dims b) Bulb A goes out and bulb B gets brighter c) Bulb B goes out and bulb A dims d) Bulb B goes out and bulb A gets brighter

Q19

A changing magnetic field is present in the interior of a circuit containing three identical resistors. Two ideal voltmeter are connected to the same points, as shown in figure. Both voltmeters show the same reading. True / False

Motional EMF

Can Magnetic field affect electric charges in any other way ? Why does a conductor moving in a Magnetic field develop polarity ? Can it be called Potential Difference ?

Q20

A metal rod moves at a constant velocity in a direction perpendicular to its length. A constant uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity. Select the correct statement(s) a) the entire rod is at the same electric potential b) there is an electric field in the rod c) the electric potential is highest at the center of the rod and decreases towards its ends d) the electric potential is lowest at the center of the rod and increases towards its ends ii) Find the EMF induced across the ends of a straight rod moving as shown in figure. iii) Find the potential difference between each pair of points marked in same color

Q21

a) A metal rod of length l rotates about its end in a plane perpendicular to a uniform magnetic field. If the angular velocity of rotation is w, find the EMF induced between the ends of the rod. b) The rod is now pivoted at a distance l1 from one end and l2 from the other end. Find the EMF induced between the ends of the rod. c) A rod of length l1 rotates about its end in a plane perpendicular to a uniform magnetic field. The rod is such that a length l2 of the rod from the pivot is made of wood and the rest is made of metal. If the angular velocity of rotation is w, find the potential difference between the ends of the rod. d) A metal rod of length l rotates about its end in a plane perpendicular to a uniform magnetic field. If the rod starts rotating with constant angular acceleration a, find the EMF induced between the ends of the rod as a function of time.

Q22

A curved wire is rotating in a plane perpendicular to a uniform magnetic field B, as shown in figure. One end of wire is the pivot and it rotates with an angular speed w. Find the induced emf between the ends of wire. Figure shows a straight conducting rod rotating with angular speed w in uniform magnetic field B. Find the induced emf between the ends of wire. A rod of length l rotates in the form of a conical pendulum with an angular velocity w about its axis as show in figure. The rod makes an angle q with the axis. What is the magnitude of the motional emf developed across the two ends of rod ? A conducting circular arc is rotating in a plane perpendicular to a uniform magnetic field B, as shown in figure. Find the induced emf between the ends of arc. A conducting ring is rotating in a plane perpendicular to a uniform magnetic field B, as shown in figure. Find the induced emf between the pivot and the points marked on ring.

EMF & Faraday\'\'s Law

Is Motional EMF due to Faraday?s Law ? Or is Faraday?s Law due to Motional EMF ? Are Motional EMF and Faraday?s Law related by a common cause ? Or are they 2 separate phenomenon ?

Q23

A rectangular metallic loop with resistance R is moved at a constant speed v across a uniform magnetic field B confined to a region of width D ( D > L ) as shown in fig. What is the external force required in each region, to move the loop with constant speed v ?

Q24

Consider the circuit shown in figure. The rod slides at constant speed v and the plane of the circuit is perpendicular to a uniform magnetic field. Till the rod is in the triangular region, a) What is the EMF induced in the rod ? b) What is the EMF induced in the circuit ?

Q25

Two geometrically identical wheels have different number of spokes connected from center to rim. (rim and the spokes have no resistance). One resistance of value R is connected between center and rim. Current in R will be a) higher in the wheel with less spokes b) higher in the wheel with more spokes c) same in both cases d) cannot be determined A metallic disc is rotating in a uniform magnetic field such that its normal vector makes and angle q with the magnetic field. Will the points on the rim of disc be at the same potential ?

Q26

In figure, a circular loop of radius r and resistance R, is centered on a long straight wire. At time t = 0, current in the long straight wire starts to increase as i = A t. The straight wire is insulated; so there is no electric contact between it and the circular loop. What is the magnitude of the current induced in the loop at time t ?

Q27

A rod of length l is rotating in a uniform magnetic field in two different orientations as shown in figure. If speed of rod in first case is v, what is the EMF induced in the rod ? If angular speed of rod in second case is w, what is the EMF induced in the rod ?

Rotating Loop

Is the induced EMF in a rotating loop due to Faraday?s Law or due to Motional EMF ?

Energy Transfers

Does Magnetic force do any work ? Then how does Magnetic Field enables the transfer of Electric energy to Mechanical energy and vice-versa ?

Q28

A conducting rod fall vertically over two vertical rails which are joined at the top through resistance R. The rails are without friction and resistance. There is a horizontal uniform magnetic field of magnitude B perpendicular to the plane of the ring and the rails and directed into the screen. What is the terminal speed acquired by rod ? What is the speed of rod as a function of time ? ( take t = 0 when the rod starts falling )

Q29

Two conducting rails with negligible resistance are laid parallel to each other, as shown in fig. to make a smooth inclined plane of inclination q with the horizontal. A conducting rod of mass m and negligible resistance is kept on the wires. A uniform magnetic field is directed in vertically upward direction. At the upper end, the wires are connected by a resistance R. Find the terminal velocity attained by rod ? Find the velocity of the rod as a function of time ?

1

Figures shows two circular rings of radii a and b (a > b) joined together with wires of negligible resistance. Both the arrangements are placed in a uniform time varying magnetic field dB/dt = K, perpendicular to the plane of the loops. Find the induced EMF in each case.

2

The figure shows certain wire segments joined together to form a coplanar loop. Resistance of wires per unit length is R. 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 ( dB/dt ). Find the magnitude and direction of current in the loop.

3

A long copper wire carries a current i. Calculate the magnetic flux through a plane surface of length l, as shown in figure. Inner edge of surface is at a distance a from the wire and outer edge is at a distance b from the wire.

4

A conducting rod of mass m is rotated with angular velocity w about its end such that its other end is connected to a metallic ring of radius r. Entire system is placed in a perpendicular uniform magnetic field B as shown in figure. The central end of the spoke is connected to the rim of the wheel through a resistor R as shown. The resistor does not rotate. What constant force F is required on the other end of rod so as to maintain constant angular velocity of the rod ? For the figure shown, a conducting diametrical rod is rotating. Resistors are not rotating. What will be the current in resistors R1 and R2 ?

5

A conducting ring of radius r and resistance R rolls on a horizontal surface with constant velocity v. The magnetic field B is uniform and is normal to the plane of the ring. What is the potential difference between the highest point and the point of contact ? Two identical conducting rings A and B of radius r are rolling over a horizontal conducting plane with same speed v but in opposite directions. A constant magnetic field B is present pointing into the plane of paper. What is the potential difference between the highest points of the two rings ?

6

Two identical circular loops of metal wire are lying horizontally on a table without touching each other. Loop A carries a current which increases with time. In response, the loop B

7

Two circular similar, coaxial loops carry equal current in the same direction. If the loops are brought nearer, a) Current will increase in both loops b) Current will decrease in both loops c) Current will remain same in both loops d) Current will increase in one and decrease in the other If another similar coaxial loop is placed in the middle of two loops, will there be any current in the center loop ?

8

A thermocol vessel contains 1 kg of distilled water at 30 oC. A metal coil of area 5 x 10-3 m2, number of turns 100, mass 0.05 kg and resistance 1 W is lying horizontally at the bottom of the vessel. A uniform time varying magnetic field is set-up to pass vertically through the coil at time t = 0. The field is first increased from 0 to 1 T at a constant rate between 0 and 0.2 and then decreased to zero from 0.2 s to 0.4 s. The cycle is repeated 10000 times. Make sketches of the current through the coil and the power dissipated in the coil as a function of time for one cycle. Assume that no heat is lost to the vessel or the surroundings. Determine the final temperature of the water under thermal equilibrium. Specific heat of metal = 500 J/kg-K and the specific heat of water = 4200 J/kg-K Neglect the inductance of coil.

9

The wire frame, shown in figure is made by taking a flat rectangular loop of sides 2a and a and bending at the long sides at their midpoints to produce two mutually perpendicular square parts. The loop is placed in an oscillating magnetic field B = Bo sin wt . The magnetic field is inclined at angle q with xy plane. For what angle q will the induced EMF in coil be maximum ?

10

A rectangular coil with its plane vertical, is released from rest in a horizontal uniform magnetic field B as shown in figure. The acceleration of the coil is a) less than g for all the time till the loop crosses the magnetic field completely b) less than g when it enters the field and greater than g when it comes out of the the field c) g all the time d) less than g when it enters and comes out of the field but equal to g when it is within the field

11

A magnetic field B = Bo y/a is acting into the screen as shown in figure. Bo and a are positive constants. A square loop of side a, mass m and resistance R lies perpendicular to Magnetic Field. It starts falling under the influence of gravity. Find: a) induced current in the loop and its direction b) net magnetic force acting on the loop and its direction c) an expression for the speed of the loop as a function of time and its terminal velocity.

12

Figure shows a rectangular conducting loop being pulled at constant speed v through two regions of uniform magnetic field. Fig gives the current i induced in the loop as a function of the position x of the right side of the loop. Take positive current to be anti-clockwise. What is the direction (into or out of the screen) of the magnetic field in region 1 ? What is the direction of the magnetic field in region 2 ? What is the ratio of magnitude of field in region 1 and 2 ?

13

Two magnetic fields exist in the two regions as shown in figure. A loop of length l and width w is placed in the field. The resistance per unit length of the loop is r. If the loop is given a velocity v towards right. What is the potential difference VA - VC ?

14

In figure shown, the rod has a resistance R and the horizontal rails have negligible friction. Battery has emf E and negligible internal resistance. The rod is released from rest. Find the velocity of the rod as function of time. What is the terminal speed of rod ? What is the current in circuit, when the rod attains its terminal speed ?

15

Set of long parallel horizontal rails, distance d apart and each having a resistance l per unit length, are joined at one end by a resistance R. A perfectly conducting rod of mass m is free to slide along the rails without friction (see figure). There is a uniform magnetic field of induction B normal to the plane of the paper and directed into the screen. A variable force Fext is applied to the rod such that, as the rod moves, a constant current i flows through R. Find the velocity of the rod and the applied force Fext as function of the distance x of the rod from R

16

A fixed conducting smooth frame lies in vertical plane. A conducting rod of mass m can move vertically and smoothly without losing contact with the frame. Rod always remains horizontal and is given an initial velocity u upwards. Taking the acceleration due to gravity as g and assuming that resistance is absent everywhere except the one shown in figure, find the time taken by rod to reach the highest point.

17

A rectangular loop with a sliding connector of length l is situated in a uniform magnetic field B perpendicular to the plane of loop. Resistance of connector is r and two resistance R1 and R2 are connected as shown in figure. Find the external force required to keep the connector moving with a constant velocity v. Repeat the above question if R2 is replaced with a capacitor with capacitance C.

18

Two parallel vertical metallic rails are separated by distance of 1 meter. They are connected at two ends by resistances R1 and R2 as shown in figure. A horizontal metallic bar of mass 0.2 kg slides down vertically without friction under the influence of gravity. There is a uniform horizontal magnetic field of 0.6 Tesla perpendicular to the plane of the rails. It is observed that when the terminal velocity is attained, the powers dissipated in R1 and R2 are 0.76 Watts and 1.2 Watts respectively. Find the terminal velocity of the bar and the values of R1 and R2.

19

A conducting wire of length l and mass m can slide without friction on two parallel rails and is connected to capacitance C. There is no resistance anywhere in the system. The whole system lines in a magnetic field B and a constant forces F is applied to the rod. Find the speed of rod at time t.

20

Consider two parallel, conducting frictionless tracks as shown in figure. A movable conductor is given a velocity u towards right. The space contains a magnetic field given by B = cx ( - k ) [ c = constant ] The mass of the conductor is m and distance between the rails is L. Find the magnetic force acting on the rod as a function of its distance from the resistor Find the heat lost during the time interval t = 0 to time t seconds, when the speed of the conductor is u/2.

21

Two long parallel conducting rails are placed in a uniform magnetic field. On one side, the rails are connected with a resistance R. Two rods are placed as shown in figure. a) If resistance of rods is zero, find the current in resistor if (i) both the rods move with speed v towards right (ii) one rod moves towards left and the other towards right (with same speed v) Repeat the above question if the rods have resistance r.

22

A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on a table. The distance between the rail is l. A conducting massless rod of resistance R can slide on the rail frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the edge of the table. A mass m is 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, find a) the terminal velocity achieved by the rod b) the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity.

23

A conducting rod is placed near an infinite current carrying wire as shown in the figure. Find the force required to move the rod with constant speed v.

24

A conducting rod is placed in a circuit as shown in figure. The system is placed in a magnetic field which is increasing at a constant rate. In which direction should the conductor move so that the net induced EMF in the circuit is zero. What external force is required to move the rod with this speed ?

25

Two identical conductors P and Q are placed on two frictionless rails in a uniform magnetic field directed into the plane. If P is moved towards right with a constant speed, then rod Q a) will be attracted towards P b) will be repelled away from P c) will remain stationary d) cannot be concluded

26

A semicircular conducting loop of radius r is rotating with constant angular velocity w about point O in the plane of screen. The effective resistance of the loop is R. a) obtain an expression for the magnitude of the induced current in the loop. b) draw a graph of induced current and the time of rotation for one full rotation. Anti-clockwise direction is taken to be positive. If the loop is set into rotation with constant angular acceleration, then c) find the variation of magnitude of induced EMF wrt time. d) draw the graph of variation of Induced EMF (with polarity) wrt time.

27

A copper rod of length L is moving with uniform velocity v parallel to a long straight wire carrying a current i. The rod is perpendicular to the wire with its closer end at distances a from it. Find the emf induced in the rod. If the rod is S - shaped, as shown in figure, find the emf induced across the ends of the rod.

28

A copper rod is bent into a semi-circle of radius r and the ends are connected to a capacitor having capacity C as shown in figure. Semi-circular section is free to rotate about axis PQ. The system is located in a uniform magnetic field of induction B such that axis of rotation PQ is perpendicular to the field direction. At initial moment of time (t = 0), plane of semi-circle is normal to the field direction and the semi circle is set in rotation with constant angular velocity w. Neglect the resistance and inductance of the circuit. Find the current flowing through the circuit as function of time.

29

A conducting rod of length l is hinged at its one end. It is free to rotate in a vertical plane. There exists a uniform magnetic field B in horizontal direction. The rod is released from the position shown in figure. What is the potential difference between the ends of the rod as a function of time ? Take friction to be zero everywhere. A conducting rod of length l is sliding along a conducting ring in vertical plane as shown in the figure. The entire arrangement is placed in a uniform magnetic field with the induction B perpendicular to the plane of the ring. The axis and the ring are connected to a current source. a) how should the current in rod vary so that the rod rotates with constant angular speed. Consider t = 0, when the rod is in its right-hand horizontal position. Consider the current to be positive when it flows from the axis of rotation towards the ring. b) What should be the emf of the source to maintain the required current? Consider the total resistance of the circuit to be constant and equal to R. Disregard the inductance of the circuit.

30

Two metallic rings A and B, identical in shape and size but having different resistivities rA and rB, are kept on top of two identical solenoids as shown in the figure. When current is switched on in both the solenoids in identical manner, the rings A and B jump to heights hA and hB, respectively, with hA > hB. The possible relation(s) between their resistivities and their masses mA and mB is (are) a) rA > rB and mA < mB b) rA < rB and mA > mB c) rA > rB and mA > mB d) rA < rB and mA < mB

31

Shown a rectangular loop of wire immersed in a non-uniform and varying magnetic field B that is perpendicular to and directed into the screen. Magnitude of field is given by B = x2 t2 , with B in teslas, t in seconds, and x in meters. The loop has width W and height H meters. What is the magnitude of the induced emf E at time t ? What is the total heat dissipated in circuit till time t ?

32

A square shaped conducting loop of side L, total mass m and total resistance R initially lies in the horizontal xy plane. There is a uniform upward magnetic field in the space within and around the loop. One side of the loop is held in place on the x - axis. When the rest of the loop is released, it begins to rotate due to the gravitational torque. a) Find the net torque that acts on the loop when it has rotated through an angle q from its original orientation and is rotating downward at an angular speed w. b) Find the angular acceleration of the loop at the instant described above c) Compared to the case with zero magnetic field, does it take the loop a longer and shorter time to rotate through 90o ? Explain. d) Is mechanical energy conserved as the loop rotates downwards?

33

Loop A of radius r moves towards loop B with a constant velocity v in such a way that their planes are always parallel. Loop B has radius R and current i. ( r << R ) Determine the magnetic flux through the smaller loop as a function of x. What is the induced emf in the smaller loop as a function of x ? What is the direction of current in loop A ?

34

A very small bar magnet of dipole moment M is pointing and moving with the speed v in the positive x-direction. A small circular conducting loop of radius a and negligible self-inductance lies in the y-z plane with its centre at x = 0, and its axis coinciding with the x-axis. Find the force opposing the motion of the magnet, if the resistance of the loop is R. Assume that the distance x of the magnet from the centre of the loop is much greater than a.

35

In the circuit shown in figure, the capacitor has capacitance C and is initially charge to Vo with the polarity shown. The resistor has resistance R. At time t = 0, the switch is closed. The smaller circuit is not connected in any way to the larger one. The wire of the smaller circuit also has a resistance R. The larger circuit is much bigger than the smaller circuit. Assume that only the wire nearest to the smaller circuit produces an appreciable magnetic field through it. Find the magnitude and direction of current in the smaller loop as a function of time. (Neglect Inductance)

36

A line charge with linear charge density l is wound around an insulating disc of mass m and radius R. Disc is placed horizontally in a uniform magnetic field Bo, pointing upwards, as shown in figure. Now the magnetic field is switched off. Find the angular speed with which the disc starts rotating.

37

A standing wave y = 2A sin kx cos wt is set up in the wire of length L, fixed at both ends by two vertical walls. The region between the walls contains a constant magnetic field B. What is the maximum induced EMF in the wire in fundamental mode ? In which modes is the EMF induced in wire always zero? What is the maximum EMF induced in wire in 3rd harmonic ?

38

A flexible wire loop in the shape of a circle has a radius that grows linearly with time ( r = a t ). There is a magnetic field perpendicular to the plane of the loop whose magnitude varies from the center of loop as : How does the emf E vary with time in each case ?

39

Two infinitely long parallel wires carrying currents I = Io sin wt in opposite directions are placed a distance 3a apart. A square loop of side a of negligible resistance with a capacitor of capacitance C is placed in the plane of wires as shown. Find the current in the square loop as a function of time.

40

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 = io sin wt flows through the solenoid. If the resistivity of the material of cylindrical shell is p, find the induced current in the shell.

41

A long circular tube of length L and radius a carries a current I ( I = Io cos wt , where Io is constant ) along its curved surface as shown. A wire-loop of resistance R and of radius r is placed inside the tube with its axis coinciding with the axis of the tube. Find the magnetic moment of the loop.

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