A current `i=alphat` , where `t` is time is flowing in a long wire. A smaller circular loop of radius a has its plane parallel to the wire and is placed at distance `d` from the wire. If resistance per unit length of loop is `lambda` , find the magnitude and direction of the current in the loop.

Explanation

Since circular loop is small, magnetic field inside loop is assumed to be same.
Magnetic field due to long wire at centre of loop
`B=(mu_(0)i)/(2pid)=(mu_(0)alphat)/(2pid),ox`
Flux through loop `phi=Bpia^(2)=(mu_(0)alphatpia^(2))/(2pid)`
Induced emf
`e=(dphi)/(dt)=(mu_(0)alphaa^(2))/(2d)`
Induced current `I=e/R=(mu_(0)alphaa^(2))/(2d.lambda.2pialpha)`
`=(mu_(0)alphaa)/(4pilambdad)`
Since `i` is increasing, flux is increasing, direction of `I` will be anticlockwise.