Electromagnetic Induction-07-PROBLEMS

PROBLEMS 1. A rectangular loop PQRS made from a uniform wire has length a, width b and mass m, it is free to rotate about the arm PQ, which remains hinged along a horizontal line taken as the y-axis (see fig- ure). Take the vertically upward direction as the z-axis. A uniform magnetic field  (3iˆ  kˆ)B exists in the region. The loop is held z P Q y B 0 a in the x-y plane and a current I is passed through it. The loop is now released and is found to stay in the horizontal position in equilib- rium. (a) what is the direction of the current I in PQ? (b) find the magnetic force on the arm RS. (c) find the expression for I in terms of B0 , a, b and m. 2. A metal bar AB can slide on two parallel thick metallic rails separated by a distance l. A resistance R and an inductance L are connected to the rails as shown in the figure. A long straight wire carrying a constant current I0 is placed in the plane of the rails and perpendicular to them as shown. The bar AB is held at rest at a distance x0 from the long wire. At t = 0, it is made to slide on the rails away from the wire. Answer the following questions : x S R b (a) Find a relation among i, di and d , where I is the current in the circuit and  is the flux of dt dt the magnetic field due to the long wire through the circuit. (b) It is observed that at time t = T, the metal bar AB is at a distance of 2x0 from the long wire and the resistance R carries a current i1 . Obtain an expression for net charge that has flown through resistance R from t = 0 to t = T. (c) The bar is suddenly stopped at time T. The current through resistance R is found to be i1 / 4 at time 2T. Find the value of L/R in terms of the other given quantities. 3. A uniform constant magnetic field → is directed at an angle of 45º to the x-axis in xy plane. PQRS is a rigid square wire frame carry- ing a steady current I0 , with its center at the origin O. At time t = 0, the frame is at rest in the position shown in the figure with its x sides parallel to x and y axes. Each side of the frame is of mass M and length L : (a) What is the torque  about O acting on the frame due to the magnetic field ? (b) Find the angle by which the frame rotates under the action of this torque in a short interval of time t , at the axis about which this rotation occurs ( t is so short that any variation in the torque during this interval may be neglected). Given : the moment of inertia of the frame about an axis through its center perpendicular to its plane is 4 ML2 . 3 4. A thermocole vessel contains 0.5 kg of distilled water at 30ºC. A metal coil of area 5 103 m2 , number of turns 100, mass 0.06 kg and resistance 1.6  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 zero to 0.8 at a constant rate between 0 and 0.2 s and then decreased to zero at the same rate between 0.2 and 0.4 s. This cycle is repeated 12000 times. Make sketches of the current through the coil and the power dissipated in the coil as functions of time for the first two cycles. Clearly indicate the magnitude of the quantities on the axes. 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 the metal  500 Jkg 1 K 1 and the specific heat of water  4200Jkg 1 K 1 . Neglect the inductance of the coil. 5. A circuit containing a two position switch S is shown in figure. (a) The switch S is in position ‘1’. Find the potential difference R5 VA  VB and the rate of production of joule heat in R1 . 1 (b) If now the switch S is put in position 2 at t  0 find 2 R (i) Steady current in R4 and (ii) The time when current in is half the steady value. Also calculate the energy stored in the inductor L at that time. 10mH