Archived Q about derivation of formula for current in AC with inductor

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

I'm reading fishbane's physics textbook right now, and in his derivation for I(t) in an AC circuit with an inductor, he does:

Vsin (wt) - L dI/dt = 0 (kirchoff's rule, V = voltage amplitude in the AC circuit, L = inductance)
then to find I, I= integral of (V/L sin(wt) dt)
which equals -V/(wL) cos (wt) + a constant

THEN he writes: "the constant must equal zero, because there is no constant emf to drive a constant current term. "

I don't understand this part. According to this then, Imax occurs at I(0) = -V/(wL). This doesn't make sense to me because when you have an inductor in the circuit, shouldn't it produce a back emf making the current slowly ramp up to a maximum?

Can someone please explain why the constant =0, and why it makes sense that Imax occurs at t=0?



The derivation assumes that the input voltage signal has been running forever, so the circuit is at steady state at time t = 0. In practical terms, imperfect materials means the resistance in the circuit may be tiny, but not zero. So any offset of current or potential that is not sustained by some constant source will be transients that "died" long ago.

You can produce the offset you describe if you suddenly start a sinewave voltage in this situation, and observe it decay over many cycles. In fact you can produce a range of "bias" offsets for the current by starting the sinusoid at different times in its cycle.

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