1. A conducting bar of resistance R slides frictionlessly with a constant speed upward over a pair of resistanceless rails inclined at an angle 2q under the action of some external force. A uniform magnetic field of strength B subsists in the region perpendicular to the plane of the rails as shown. Find the current induced in the circuit both by direct use of Faraday’s law and by explicit use of Lorentz force for calculation of motional emf.

From geometry, we have, where 2w is the instantaneous length of the bar enclosed by the circuit and y the distance from the junction of the rails. The area swept by the bar is

The emf, as given by Faraday’s law is

assuming that the bar started sliding at t=0 from the junction. The current is obtained by dividing this expression by R.

Since only the bar is moving, the motional emf is only along the sliding bar and is given by , which gives the same result as above.

2. A rectangular conducting loop of width w , length L and resistance R falls under gravity keeping its plane always vertical . An inhomogeneous magnetic field points into the page (xy) plane). Find the terminal velocity of the loop.

3. A circular loop of radius R rotates with an angular speed in a uniform magnetic field directed along the x axis. The loop rotates about an axis parallel to the z axis. Calculate the emf induced in the loop.

If is the instantaneous angle between the normal to the loop and the direction of the magnetic field,

4. A long coaxial cable consists of a thin wire surrounded by a concentric shell of radius R. Current goes up through the wire and returns through the shell. If the current varies with time at a constant rate , calculate the emf developed and the induced electric field in the rectangular loop of side shown in the figure.

We assume that the instantaneous field is given by the corresponding steady state expression, I.e. the magnetic field at a distance r for a long straight wire is given by . Thus the flux through a strip at a distance r from the wire between the wire and the she shell is so that the total flux is (since the field outside is zero),

Since the electric field can only depend on the distance,

which gives,