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, magnetic flux `phi_(B)=3(at^(3)-bt^(2))T-m^(2)`
where, `a=2s^(-3),b=6s^(-2)`
`therefore"Induced emf,"|e|=|(dphi_(B))/(dt)|=9at^(2)-6bt`
`therefore"Induced currect, i"=(|e|)/(R)=(9at^(2)-6bt)/(3)=3at^(2)-2bt`
For current to be maximum, `(di)/(dt)=0`
`therefore" "6at-2b=0 or t=(b)/(3a)`
i.e., at `t=(b)/(3a)`, current is maximum.
Substituting the values in Eq. (b), the maximum current is
`i_(max)=3a((b)/(3a))^(2)-2b((b)/(3a))`
`=(b^(2))/(3a)-(2b^(2))/(3a)=(-b^(2))/(3a)`.
Magnitude of this maximum current will be `i_(max)=(b^(2))/(3a)`.
Substituting the given value of a and b, we have
`i_(max)=((6)^(2))/(3(2))=6.0A`