system
Given the following post, respond as if you're commenting on Reddit:
human
Does the imaginary part of the voltage in an RLC circuit really exist, or is it just an artifact of the math we use to derive it that can be discarded? Is there any way to measure it?

So in my physics class that I took last semester we learned about RLC circuits and my teacher sort of handwaved away the imaginary part of the voltage, saying "we only care about the real part." Is this totally true or is there just a deeper rabbit hole he didn't want to get into for the purposes of our class?

If we have a capacitor, resistor, and inductor in series and in a closed loop with no external voltage source (for the purposes of this explanation, anyways.) We can use kirchoff's law to derive the fact that the voltage drop over any closed loop is zero, so we get:

&nbsp;

-L(dI/dt)-IR-(R/C)=0

&nbsp;

=-L(d^2 Q/dt^2 )-R(dQ/dt)-(Q/C)=0

&nbsp;

=(d^2 Q/dt^2 )+(R/L)(dQ/dt)+(1/LC)Q=0

&nbsp;

So now we have a differential equation, whose solution we can guess is something like : Q=Ae^iωt 

&nbsp;

Therefore:

&nbsp;

-ω^2 A^eiωt +iω(R/L)A^eiωt -(1/LC)A^eiωt =0

&nbsp;

=-ω^2 +iω(R/L)-(1/LC)=0

&nbsp;

Where (1/LC) is some resonant frequency ω*_0_* squared, so we can solve the quadratic:

&nbsp;

-ω^2 +iω(R/L)-ω*_0_*^2 =0

&nbsp;

to find ω, which will be:

&nbsp;

ω=i(R/L)±sqrt(ω*_0_*^2 -(R^2 /4L^2 ))

&nbsp;

[edit: i skipped some parts here, i'm just going off what i have in my notebook]

&nbsp;

so finally Q=Ae^((-R/2L)^)t ^+iωt 

&nbsp;

which is e to a complex power, which can be re-written using euler's formula, which gives a real and an imaginary part:

&nbsp;

Q=Ae^((-R/2L)^)t cos(ωt)+Ae^((-R/2L)^)t isin(ωt)

&nbsp;

so now for Q we have a real and imaginary part. So what does the imaginary part mean in reality? If we tried to see it on an oscilliscope, per se, could we? Could we have some sort of device that would measure charge or voltage in the complex plane? Or does none of this matter because the imaginary part is just an artifact of the mathematics that doesn't have any real meaning?

&nbsp;

thanks for your time, anyone who reads and is able to respond to this.
gpt
The imaginary part helps to determine the phase of the oscillatory part. It has a real meaning in the sense that different initial configurations of the capacitor and current will result in different values for the real and imaginary parts.