Electromagnetic Induction Brain-Storming Problems

1) A long solenoid of radius a and n number of turns per unit length is enclosed by a cylindrical shell of radius R, thickness d(d<<R) and length L. A variable current I = i Sin(ωt) flows through the solenoid. If resistivity of the material of the cylindrical shell is ρ, find the induced current in the shell.

2) A pair of parallel horizontal conducting rails of negligible resistance shortened at one end is fixed on a table. The distance between the rails is L. A conducting mass-less rod of resistance R can slide on the rails frictionlessly. The rod is tied to a mass-less string which passes over a pulley fixed to the edge of the table. A mass m, 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, calculate:

• the terminal velocity achieved by the rod and
• the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity.

3) A very small circular loop of area 0.0005 sq m, resistance 2Ω and negligible inductance is initially coplanar and concentric with a much larger fixed circular loop of radius 0.1 m. A constant current of 1A is passed in the bigger loop and the smaller loop is rotated with angular velocity of ω rad/s about a diameter. Calculate:

• the flux linked with the smaller loop;
• induced emf and
• induced current in the smaller loop, as a function of time.

4) Does Faraday's induced emf depend on the resistance of the material of the coil?

5) A plane conducting circular loop of area 1 sq m and 200 turns whose resistance is 1.2Ω is situated in a region of constant external magnetic field of 0.6T parallel to the axis. The loop is removed from the field region in 0.001 s. Calculate the total work done.

6) Early in 1981, the Francis Bitter National Magnet Laboratory at  M.I.T. commenced operation of a 3.3 cm diameter cylindrical magnet that produces a 30T field, then the world's largest steady-state field. The field can be varied sinusoidally between the limits of 29.6T and 30.0T at a frequency of 15Hz. When this is done, what is the maximum value of the induced electric field at a radial distance of 1.6cm from the axis? [description of this magnet can be found in the August issue of Physics Today, 1984]

7) A uniform magnetic field B is changing in magnitude at a constant rate of (dB/dt). You are given a mass m of copper that is to be drawn in to a wire of radius r and formed into a circular loop of radius R. Show that the induced current in the loop does not depend in the size of the wire or the loop and, assuming B perpendicular to the loop, is given by 

i = [m/(4πρδ)].[dB/dt]

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