A magnetic flux through a stationary loop with a resistance `R` varies during the time interval `tau` as `phi=at(tau-t)`. Find the amount of the generated in the loop during that time

Explanation

`phi=at(tau-t)=at tau-at^(2)`
`e=-(dphi)/(dt)=-atau+2at=a(2t-tau)`
`H=int_(0)^(tau)(e^(2))/(R)dt=(a^(2))/(R)int_(0)^(tau)(2t-tau)^(2)dt`
`=(a^(2))/(R)int_(0)^(tau)(4t^(2)-4taut+tau^(2))dt`
`(a^(2))/(R)|(4r^(3))/(3)-(4taut^(2))/(2)+tau^(2)T|_(0)^tau`
`=a^(2)/(R)((4tau^(3))/(3)-2tau.tau^(2)+tau^(3))`
`=(a^(2))/(R).(tau^(3))/(3)=(a^(2)tau^(3))/(3R)`