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Find the magnetic field B(s,t) produced for s<R (inside solenoid), s>R (outside solenoid), and the associated electric fields.

I had first considered using Ampere's Law to find the magnetic field, but realized that since the current is not constant that I can't use it. But I'm not sure if this works with time-dependence, since it is technically still a constant in terms of position (x,y,z). Can Ampere's Law be applied here? Otherwise I don't really know how to approach the problem since the Biot-Savart Law seems to be too complicated. Also, shouldn't the magnetic field outside the solenoid be zero?

And once I find out B, do I have to apply Maxwell's equations to find E?

I'm sorry I don't have an elaborate attempt, I'm really pretty stuck on this question.