Magnetics-06-SUBJECTIVE UNSOLEVED LEVEL - II

SUBJECTIVE UNSOLEVED LEVEL - II (BRUSH UP YOUR CONCEPTS) y 1. A wire carrying a 12 A current is bent to pass through various 2 corners of a square of side 20 cm as shown in the figure. What is 1 ˆ 3 the force on each segment for B  0.5 kT . x 4 z x 2. Ina certain region uniform electric field E and magnetic field B are present in the opposite direction. At the instant t = 0, a particle of mass m carrying a charge q is given velocity v0 at an angle  , with the y-axis, in the yz plane. Find the time after which the speed of E B z the particle would be minimum.  y v0 3. A particle of mass m having a charge q enters into a circular region of radius R with velocity v directed towards the centre. The strength of magnetic field is B. Find the deviation in the path of the particle. O 4. Figure shows a rod PQ of length 20.0 cm and mass 200 g sus- pended through a fixed point O by two threads of lengths 20.0 cm each. A magnetic field of strength 0.500 T exists in the vicinity of the wire PQ as shown in the figure. The wires connecting PQ with the battery are loose and exert no force on PQ. (a) Find the ten- sion in the threads when the switch S is open. (b) A current of 2.0 A is established when the switch S is closed. Find the tension in the threads now. 5. A particle of mass m and positive charge q, moving with a uniform v  velocity v, enters a magnetic field B as shown in figure. (a) Find the radius of the circular arc it describes in the magnetic field, (b) Find the angle subtended by the arc at the centre, (c) How long does the particle stay inside the magnetic field? (d) Solve the three parts of the above problem if the charge q on the particle is negative. × × × × × × ×B × × × × × 6. A tightly-wound, long solenoid has n turns per unit length, a radius r and carries a current i. A particle having charge Q and mass m is projected from a point on the axis in a direction perpendicular to the axis. What can be the maximum speed for which the particle does not strike the solenoid ? 7. A 1.15 kg copper rod rests on two horizontal rails 95.0 cm apart and carries a current of 53.2 A from one rail to the other. The coefficient of static friction is 0.58. Find the smallest magnetic field (not necessary vertical) that would cause the bar to slide. 8. A beam of electrons whose kinetic energy is K emerges from a thin-foil “window” at the end of an accelerator tube. There is a metal plate a distance d from this window and at right angles to the direction of the emerging beam (see figure). (a) Show that we can prevent the beam from hitting the plate if we apply a magnetic field B such that B  in which m and e are electron mass and charge. (b) How should B be oriented ? 9. A deuteron in a cyclotron is moving in a magnetic field with an orbit radius of 50 cm. Because of a grazing collision with a target, the deuteron breaks up, with a negligible loss of kinetic energy, into a proton and a neutron. Discuss the subsequent motions of each. Assume that the deuteron energy is shared equally by the proton and neutron at breakup. 10. Two long, straight, parallel wires, separated by 0.75 cm, are perpendicular to the plane W1 of the page as shown in figure. Wire W1 carries a current of 6.6 A into the page. What must be the current (magnitude and direction) in wire W2 for the resultant magnetic W2 field at point P to be zero ? P 0.75cm 1.5cm SUBJECTIVE UNSOLVED LEVEL - III (CHECK YOUR SKILLS) 1. Figure shows a cross-section of a long, cylindrical conductor of radius R containing a long, cylindrical hole of radius a. The axes of the two cylinders are parallel and are a distance b apart. A current i is uniformly distributed over the shaded area in the figure. (a) Use superposition ideas to show that the magnetic field at the center of the hole is B  0ib 2(R 2  a 2 ) (b) Discuss the two special cases a = 0 and b = 0. (c) Can you use Ampere’s law to show that the magnetic field in the hole is uniform? (Hint : Regard the cylindrical hole as filled with two equal currents moving in opposite directions, thus canceling each other. Assume that each of these currents has the same current density as that in the actual conductor. Thus we superimpose the fields due to two complete cylinders of current, of radii R and a, each cylinder having the same current density). 2. A particle having mass m and charge q is released from the origin in a region in which electric field and magnetic field are given by → ˆ → ˆ B  B0 j and E  E0k Find the speed of the particle as a function of its z-coordinate. 3. A non-conducting thin spherical shell of radius R has uniform surface charge density  . The cell rotates about a diameter with constant angular velocity  . Calculate (a) magnetic induction B at the centre of the cell. (b) magnetic moment of the sphere. 4. Two particles, each having a mss m are placed at a sepa- ration d in a uniform magnetic field B as shown in figure. They have opposite charges of equal magnitude q. At time × × × × q × v × × × × × × × × × × -q × × × d t = 0, the particles are projected towards each other, each with a speed v. Suppose the Coulomb force between the charges is negligible in comparision to magnetic force. × × × × ×B × (a) Find the maximum value vm of the projection speed so that the two particles do not collide. (b) What would be the minimum and maximum separation between the particles if v = vm/2? (c) At what instant will a collision occur between the particles if v = 2vm? (d) Suppose v = 2vm and the collision between the particle is completely inelastic. Describe the motion after the collision. 5. Consider a particle of mass m and charge q moving in the xy plane under the influence of a uniform → magnetic field B pointing in the +z direction. Write expressions for the coordinates x(t) and y(t) of the particle as functions of time t, assuming that the particle moves in a circle of radius R centered at the origin of coordinates, in anticlockwise sense and its radius vector makes and angle θ0 with the x-axis at time t  0 . 6. Figure shows a wire ring of radius a at right angles to the gen- eral direction of a radially symmetric diverging magnetic field. The magnetic field at the ring is everywhere of the same mag- nitude B, and its direction at the ring is everywhere at an angle  with a normal to the plane of the ring. The twisted lead wires have no effect on the problem. Find the magnitude and direc- tion of the force the field exerts on the ring if the ring carries a current i as shown in the figure. 7. Figure shows a wooden cylinder with a mass m = 262 kg and a length L = 12.7 cm, with N = 13 turns of wire wrapped around it longitudinally, so that the plane of the wire loop contains the axis of the cylinder. What is the least current through the loop that will prevent the cylinder from rolling down a plane inclined at an angle  to the horizontal, in the presence of a vertical, uniform magnetic field of 477 mT, if the plane of the windings is parallel to the inclined plane ? 8. Ina certain region of space there exists a uniform and constant electric field of magnitude E along the positive y-axis of a co-ordinate system. A charged particle of mass ‘m’ and charge ‘-q’ (q > 0) is projected with speed 2v at an angle of 60º with the positive x-axis in x-y plane from the origin. When the x-coordinate of the particle becomes 3mv2 , a uniform and constant magnetic field of strength qE B is also switched on along the positive y-axis. Find the co-ordinate of the particle as a function of time t after its projection. 9. A long straight metal rod has a very long hole of radius a drilled parallel to rod axis at a distance c from the axis of the rod as shown in the figure. If the rod carries a current i. Find the magnetic field (a) on the axis of the rod. (b) on the axis of the hole. 10. A loop, carrying a current I, lying in the plane of the paper, is in the field of a long straight wire with current 0 (inwards) as shown in the figure. Find the torque acting on the loop.