#### Exercise

**Question 1. Predict the direction of induced current in the situations described by the following (a) to (f).**

**Sol.**

**(a)**Apply Lenz’s law, induced current will flow to oppose the magnet. So South-pole is developed at end q and current flows in qrpq direction.**(b)**Same as above, South-pole developed at q and current is in prqp direction. For coil 2, North- is going away so attractive i.e., South pole is developed at end x and direction is yzxy.**(c)**As key is closed, magnetic flux rises in the first coil, in other coil, induced current would be such as to oppose this increasing magnetic flux. This happenes when magnetic field produced in this coil is from right to left hence current is in yzx direction.**(d)**As rheostat is changed to decrease resistance, current will increase and magnetic flux linked with it will rise. Applying Lenz’s law, the induced current would oppose this increase, hence direction of magnetic field should be right to left and current is in direction zyx.**(e)**When current is flowing before release of tapping key magnetic flux is from left to right (right end is N-pole). As key is released, current decreases magnetic flux decreases, hence induced flux would oppose this decrease and favour an increase in flux. Hence direction ofinduced current in adjoining coil is from left to right in direction xry.**(f)**As current decreases in straight conductor, magnetic field in the plane of the coil remains to be in the same plane. The circular coil does not see the change in magnetic field (same plane) hence no current (or flux) is induced.

**Note: **Use Lenz ‘slaw to find the direction of induced current

**Question 2. Use Lenz’s law to determine the direction of induced current in the situations described by Fig.(a) A wire of irregular shape turning into a circular shape;(b) A circular loop being deformed into a narrow straight wire.**

**Sol**.**(a)** As wire increases its area, the magnetic flux linked with the loop increases, the induced emf causes a current to oppose it. The force should be inward (to pull the wire back) hence direction of current is adcba

**(b)** Here, the area decreases, hence current induced opposes this decrease in field and would try to increase the flux by pulling wire outward hence

current is in direction a ‘d ‘c’ b ‘a’.

**Question 3. A long solenoid with 15 turns per cm has a small loop of area 2.0 cm2 placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s, what is the induced emf in the loop while the current is changing?**

**Note:** Refer Chapter at a Glance (2)

**Question 4. A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the loop. (a) What is the emf developed across the cut if the velocity of the loop is 1 cm s-1 in a direction normal to the (1) longer side, (2) shorter side of the loop? (b) For how long does the induced voltage last in each case?**

**Note:** Emf produced due to motion of a conductor in a magnetic field is called motional emf.

**Question 5. A 1.0 m long metallic rod is rotated with an angular frequency of 400 rad s-1 about an axis normal to the rod passing through its one end. The other end of the rod is in contact with circular metallic ring. A constant and uniform magnetic field of 0.5 T parallel to the axis exists everywhere. Calculate the emf developed between the centre and the ring.**

**Question 6. A circular coil of radius 8.0 cm and 20 turns is rotated about its vertical diameter with an angular speed of 50 rad s-1 in a uniform horizontal magnetic field of magnitude 3.0 x 10-2 T. Obtain the maximum and average emf induced in the coil. If the coil forms a closed loop of resistance 10 fl, calculate the maximum value of current in the coil. Calculate the average power loss due to Joule heating. Where does this power come from?**

**Question 7. Horizontal straight wire 10 m long extending from east to west is fallingwith a speed of 5.0ms-1, at right angles to the horizontal component of the earth’s magnetic field, 0.30 x 10-4 Wb m-2.**

**(a) What is the instantaneous value of the emf induced in the wire?****(b) What is the direction of the emf?****(c) Which end of the wire is at the higher electrical potential?**

**Question 8. Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V induced , give an estimate of the self-inductance of the circuit.**

**Question 9. A pair of adjacent coils has a mutual inductance ofl.5 H. If the currentin one coil changes from 0 to 20 A in 0.5 s, what is the change of flux linkage with the other coil?**

**Question 10. A jet plane is travelling towards west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing having a span of 25 m, if the Earth’s magnetic field at the location has a magnitude of 5 x 10-4 T and the dip angle is 30°.**

**Note:** Refer Chapter at a Glance (3)

##### ADDITIONAL NCERT EXERCISES

**Question 11. Suppose the loop shown in figure is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that field decreases from its initial value of 0.3 T at the rate of 0.02 Ts-1• If the cut is joined and the loop has a resistance of 1.6 ohm, how much power is dissipated by the loop as heat? What is the source of this power?**

**Question 12. A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity of 8 cm s-1 in the positive x-direction in an environment containing a magnetic field in the positive z-direction. The field is neither uniform in space nor constant in time. It has a gradient ono-3 Tcm-1 along the negative x-direction (that is it increases by 10-3 T cm-1 as one moves in the negative x-direction). and it is decreasing in time at the rate of 10-3 Ts-1• Determine the direction and magnitude of the induced current in the loop if its resistance is 4.50 mQ.**

**Note: **Refer Chapter at a Glance (1)**Note:** Magnetic field have is not constant in space and is also changing with time. The induced emf has contribution from both.

**Question 13. It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. Asmall flat search coil of area 2 cm2 with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it quick 90° turns to bring its plane parallel to the field direction. The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is7.5 mC. The combined resistance of coil and the galvanometer is 0.50 Q. Estimate the field strength of magnet.**

**Question 14. Figure shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod= 15cm, B = 0.50 T, resistance of the closed loop containing the rod= 9.0 m Q. Assume the field to be uniform.**

**Suppose K is open and the rod is moved with a speed of 12 cm s-1 in the direction shown. Give the polarity and magnitude of the induced emf.(b) Is there an excess charges built up at the ends of the rods when K is open? What ifK is closed?(c) With K open and the rod moving uniformly, there is no net force on the electrons in the rod PQ even though they do experience magnetic force due to the motion of the rod Explain.(d) What is the retarding force on the rod when K is closed?(e) How much power is required (by an external agent) to keep the rod moving at the same speed (=12 cm s-1) when K is closed?How much power is required when K is open?(f) How much power is dissipated as heat in the closed circuit? What is the source of this power?(g) What is the induced emf in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?**

**Question 15. An air cored solenoid with length 30 cm, area of cross-section 25 cm2 and number of turns 500, carries a current of 2.5 A, The current is suddenly switched off in a brief time of 10-3 s. How much is the average back emf induced across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid.**

**Question 16. (a) Obtain an expression for mutual inductance between a long straight wire and a square loop of side a as shown in figure.**

**(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity v = 10 ms-•. Calculate the induced emf in the loop at the instant when x = 0.2 m. Take a= 0.lm and assume that the loop has a large resistance.**

**Note:** Magnetic field does not remain constant throughout the strip. it shall be required to find the elementry flux.

**Question 17. A line charge A per unit length is lodged uniformly onto the rim of a wheel of mass Mand radius R. The wheel has light non-conducting spokes and is free to rotate without friction about its axis. A uniform magnetic field extends over a circular region within the rim. It is given by**

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