Question

A wire of length l in the form of a square loop lies in a plane normal to a magnetic field `B_(0)`. If this wire is converted into a circular loop in time, then find the average induced emf.

Answer 1

Total length of circle will be l = `2pirrArrr=(l)/(2pi)`
Side of square, `L=(l)/(4)`.
Using the relation `phi_(B)=B.S=BS cos theta`
Here, `theta=0`
`rArr" "phi_(B)=BS`
Magnetic flux of square loop will be
`phi_(1)=B_(0)L^(2)=(B_(0)l^(2))/(16)`
Magnetic flux of circular loop will be
`phi_(2)=B_(0)pir^(2)=(B_(0)pil^(2))/(4pi)=(B_(0)l^(2))/(4pi)`
Average induced emf is given by
`bare=-(phi_(2)-phi_(1))/(Deltat)=-(B_(0)l^(2))/(t_(0))((1)/(4pi)-(1)/(16))=(B_(0)l^(2))/(16pit_(0))(4-pi)`