Expression for current through loop as a function of time

1. The problem statement, all variables and given/known data

An N-turn circular coil of radius r with a total resistance of R is placed such that the normal to its plane is parallel to the +z axis. A uniform magnetic field varies with time according to B=B0sin([itex]\omega[/itex]t) where the amplitude B0 and angular frequency ω are constants. Find an expression for the current in the loop as a function of time.

2. Relevant equations

1. [itex]\Phi[/itex]B=[itex]\int[/itex]BcosA
2. ε=d/dt [itex]\Phi[/itex]B
3. ε=iR

3. The attempt at a solution
I took the integral of the magnetic field to find flux (used equation 1 from above). I multiplied by N to account for the N number of loops. Then I took the derivative of the magnetic flux (used equation 2) to determine ε. After, I just plugged in ε into the 3rd equation and solved for current… But it doesn’t really make sense to plus in a value that varies with time into an equation that deals only with constant values.

I think I missed something important here, so any help would be appreciated. I’ve also included an attachment of the question and the answer I got using the steps I outlined above.


Attached Images
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