Q24 of 31 Page 170

A uniform magnetic field of 3000 G is established along the positive z-direction. A rectangular loop of sides 10 cm and 5 cm carries a current of 12 A. What is the torque on the loop in the different cases shown in Fig. 4.28? What is the force on each case? Which case corresponds to stable equilibrium?

Given:


Magnetic field strength, B = 3000G


B = 3000 × 10-4T


B = 0.3 T


Length of rectangular loop = 10 cm


Width of loop = 5 cm


Current in loop = 12 A


We know that,


Torque, T = IA × B …(1)


Where, I = current through loop


A = area of cross-section


B = magnetic field strength


Note: We take vector normal to the cross-section for A.


For each of the case we plug in the variables in equation (1).


(a) T = 12A × (50 × 10-4) î m2 × 0.3 T


= -1.8 × 10-2 ĵ Nm (∵ î × = -ĵ )


Force on the loop is Zero as the angle between area of cross section and magnetic field is Zero.


(b) This case is similar to A, hence the results will also be same. There is no change in any vectors.


(c) T = IA × B


We observe that A is perpendicular to x-z plane and B is along z axis.


⇒ T = -12A × (50 × 10-4)m2 ĵ × 0.3T


⇒ T = -1.8 × 10-2 î Nm (∵ ĵ × = -î )


(d) Magnitude of torque,


|T| = IA × B


⇒ |T| = 12 × (50 × 10-4)m2 × 0.3T


⇒ |T| = 1.8 × 10-2 Nm


This torque is makes 240° with the positive x direction. The force again is Zero.


(e) T = IA × B


= 12 × (50 × 10-4) m2 × 0.3 T


= 0


Since cross section and magnetic field are in same direction net torque is zero. Net force is also Zero.


(f)


= 12 × (50 × 10-4) × 0.3


= 0


Hence the torque and forces are zero.


Stable equilibrium:


In case E, the angle between A and B is zero. If we displace the wire, it will come back in this position, hence it is the stable equilibrium condition.


Unstable equilibrium:


In case F, the angle between A and B is 180°. If the wire is displaced, it will not come back in this position so we can conclude that it is the case for unstable equilibrium.


More from this chapter

All 31 →
22

The wires which connect the battery of an automobile to its starting motor carry a current of 300 A (for a short time). What is the force per unit length between the wires if they are 70 cm long and 1.5 cm apart? Is the force attractive or repulsive?

 23

A uniform magnetic field of 1.5 T exists in a cylindrical region of radius 10.0 cm, its direction parallel to the axis along east to west. A wire carrying current of 7.0 A in the north to south direction passes through this region. What is the magnitude and direction of the force on the wire if,

(a) the wire intersects the axis,


(b) the wire is turned from N-S to northeast-northwest direction,


(c) the wire in the N-S direction is lowered from the axis by a distance of 6.0 cm?

 25

A circular coil of 20 turns and radius 10 cm is placed in a uniform magnetic field of 0.10 T normal to the plane of the coil. If the current in the coil is 5.0 A, what is the

(a) total torque on the coil,


(b) total force on the coil,


(c) average force on each electron in the coil due to the magnetic field?


(The coil is made of copper wire of cross-sectional area 10–5 m2, and the free electron density in copper is given to be about 1029 m–3.)

 26

A solenoid 60 cm long and of radius 4.0 cm has 3 layers of windings of 300 turns each. A 2.0 cm long wire of mass 2.5 g lies inside the solenoid (near its centre) normal to its axis; both the wire and the axis of the solenoid are in the horizontal plane. The wire is connected through two leads parallel to the axis of the solenoid to an external battery which supplies a current of 6.0 A in the wire. What value of current (with appropriate sense of circulation) in the windings of the solenoid can support the weight of the wire? g = 9.8 m s–2.