(a) What is the maximum value of the current in the circuit? I max =_______A (b)

Question

(a) What is the maximum value of the current in the circuit?

Imax =_______A

(b) What are the maximum values of the potential difference across the resistor and the inductor?
?VR,max =________V

?VL,max =________V

(c) When the current is at a maximum, what are the magnitudes of the potential differences across the resistor, the inductor, and the AC source?
?VR = _________V

?VL =_________V

?Vsource =_________V

(d) When the current is zero, what are the magnitudes of the potential difference across the resistor, the inductor, and the AC source?
?VR =_________V

?VL =_________V

?Vsource =_________V

Explanation / Answer

(a) Of course, the answer would be V/R if the current was static and in steady state. But alternating the current activates an opposing voltage in the coil/inductor to resist the current:
V = - L (d/dt) i

The negative means V is in one direction while the current i inducing the V is in the other direction. So in the inductance circuit charge is analogous to mass in the simple harmonic oscillator, in that there's an inertial component that dampens acceleration (of the charges). So resistors, inductors, and capacitors all share the property of slowing the current in some way (in an AC circuit).

In fact, even if the power source was DC, the current wouldn't jump up to maximal value immediately after the DC source was turned on. It would approach it asymptotically over time, because the countering voltage from the inductor would oppose the current for awhile. So this is not a strictly AC phenomenon.

But the AC circuit won't have time to reach this maximum current, 170/1.2, before the voltage reverses. So we know what the maximum current is NOT.

So while R works like air resistance, dissipating energy away, L works to store up energy, in its magnetic field, sorta like a harmonic oscillator stores up KE from the overshooting mass as PE.

We have these two solution forms:
applied alternating EMF = maximum applied EMF * sin (2? freq * t)
actual, circuitwide current = i * sin (2? freq * t - phi)

for the differential equation
applied alternating EMF = voltage drop across resistor + voltage drop across inductor

(There's a rate in the right-hand side terms, so it's a diff. eqn.)

Note that the current is the same everywhere, by conservation of mass, i.e. by Kirchoff's first law.

The goal is to find phi and i.

You can solve the DE, or by phasors and the Pythagorean theorem, as is done in the slides of the second link below.

However it's solved, the solution is
amplitude of current = i= maximum applied EMF / square root [R^2 + (2? freq L)^2]

Note that what's meant by "amplitude" is that that's the largest value it attains.

(b) For the resistor, you just use the same equation as always,
potential drop = current * resistance

For the inductor, again use the same equation as always,
potential drop = L di/dt
potential drop = current * 2? freq * L
In other words, the 2? freq term comes out of the trig function when you take the first derivative with respect to time.

Just plug in the maximum current possible, found in (a).

If the nature of the voltage drop across the inductor is hard to get used to, just consider that if the inductor wasn't there, there wouldn't be a voltage drop. Also, if the power source was DC, then at t->?, there would be no voltage drop across it either. The voltage drop comes from the opposing EMF the coil sets up, driven by the change in current coming from the AC source. It reduces the potential as if it's a sort of current-driven resistor. If the current doesn't change, it doesn't resist, and there's no potential drop across it.

(c) tan phi = (XsubL - XsubC) / R
where
XsubL = inductive reactance = 2? freq * L
XsubC = capacitive reactance = 1 / (2? freq * C)
freq = frequency of applied emf

Now with the value for phi, you can return to the equation for the actual, circuitwide current, back in (a), then solve for t that maximizes current. Use the corresponding voltage drop equations for the resistor and the inductor (see the equations on slide 39 of the second link):
VsubR = i R sin (2? freq t)
VsubL = 2? freq L i cos (2? freq t)

(d) Same strategy as (c).

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