1.

Conductor of length l has shape of a semi cylinder of radius R(ltlt l). Cross section of the conductor is shown in the figure. Thickness of the conductor is (ltlt R) and conductivity of its material varies with angle theta according to the law sigma = sigma_(0)cos theta where sigma_(0) is a constant. If a battery of emf epsilon is connected across its end faces (across the semi-circular cross-sections), the magnetic induction at the mid point O of the axis of the semi-cylinder is :

Answer»

`(mu_(0)sigma_(0)epsilon t)/(8L)`
`(mu_(0)sigma_(0)epsilon t)/(4 l)`
`(mu_(0)sigma_(0)epsilon t)/(l)`
`(2mu_(0)sigma_(0)epsilon t)/(l)`

Solution :
`dB=(mu_(0)di)/(2piR)=(mu_(0)xx(epsilon)/(1)(sigma_(0)cos theta)Rd theta XXT)/(2piR)`
`B=int_(0)^(pi//2)2dBcostheta=(mu_(0)sigma_(0)epsilon t)/(4l)`


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