The magnetic flux passing through a metal ring varies with time `t` as: `phi_(B)=3(at^3-bt^2)T-m^2` with `a=2.00s^-3` and `b=6.00s^-2`. The resistance of the ring is `3.0Omega`. Determine the maximum current induced in the ring during te interval from `t=0` to `t=2.0s`.

Explanation

Given `phi_B=3(at^3-bt^2)`
`:. |e|=|(dphi_B)/(dt)|=9at^2-6bt`
`:.` Induced current, `i=(|e|)/R=(9at^2-6bt)/3=3at^2-2bt`
For cuurrent to be maximum,
`(di)/(dt)=0`
`:. 6at-2b=0`
or `t=b/(3a)`
i.e. at `t=b/(3a)`, current is maximum. This maximum current is
`i_max.=3a(b/(3a))^2-2b(b/(3a))`
`= b^2/(3a)-(2b^2)/(3a)=b^2/(3a)`
Substituting the given values of `a` and `b` we have
`i_max=((6)^2)/(3(2))=6.0A`