A wire whose cross-sectional area is and whose resistivity is is bent into a circular arc of radius as shown in the figure. An additional straight length of this wire, OP, is free to pivot about O and makes sliding contact with the arc at P. Finally, another straight length of this wire, OQ, completes the circuit. The entire arrangement is located in a magnetic field directed out of the plane of the figure. The straight wire OP starts from rest with and has a constant angular acceleration of . (a) We have to find the resistance of the loop OPQO as a function of . (b) We have to find the magnetic flux through the loop as a function of . (c) We have to find the value of the angle for which induced current in the loop is a maximum. (d) We have to find the maximum value of the induced current in the loop.

Solution:             Click For PDF Version

(a) and (b)

As the wire OP is moving with constant acceleration,

,

and OP starts from rest with , the change with time of angle will be given by the function

.

The flux enclosed by the loop OQPO will, therefore, be

By the Faraday’s law of induction with the change in flux in the loop emf will get developed, which is given by

The resistance of the loop will change with time. The length of the arc PQ is changing because P is sliding along the circumference of the semicircle. The resistance of the arc will be given by the function

where is the resistivity and a is the cross-sectional area of the wire.

(c) and (d)

Induced current in the loop as a function of will be given by the expression

For finding the maximum value of we will calculate the extremum of the function .

The maximum value of the induced current will therefore be