Here is an example of electromagnetic induction and Faraday's law for a constant B-field and a changing area. A conducting hoop/circuit moves into a B-field, and we determine the induced voltage (i.e. emf) and current. I'll make mention of two different magnetic forces that are relevant here: first, F = qv x B is the force that physically gets the current started since a conductor with free charges is moving through a B-field; second, once that current is turned on, F = Il x B turns on to try and slow the circuit down (magnetic brake). Lenz's law is also discussed.
One other aspect of this is the determination of the velocity of the circuit as a function of time. The magnetic braking force is analyzed with Newton's 2nd law, and we get a similar result as we did in mechanics with air friction, where the force is exponential in time. I hope this helps!