Now, this is worth a little bit of discussion.

It looks like we, specialized our current voltage to be something very specific,

just a sinusoidally varying function. Now the point here, and we'll talk more

about this closer to the end of the course.

The point here is we really have not lost any generality at all.

We haven't specified the frequency. This can be any frequency that you want,

from zero to infinity. And there's something that we're going to

look at later on called Fourier analysis, which says that you can construct any

arbitrary function by adding together appropriate combinations of sine and

cosine functions. And so, if we're able to analyze a

circuit for one frequency or in more to the point for any frequency.

Omega is unspecified, then we can figure out what that circuit response is going

to be in response to any arbitrary function.

Because the this is something very important.

All of the circuits that we're looking at here are linear, and I can add together

the response of the circuit at every frequency and I can add those together

these various frequencies together to create any arbitrary function of time.

So, solving for the response of a circuit, at any old frequency, omega, an

unspecified frequency, enables us to ultimately figure out the response of

that circuit to any arbitrarily complicated function of time.

So, for now, all we have to do is worry about what's the response of a circuit

for a sinusoidal input at some frequency omega, where omega is general.

It can be any frequency that you want. So we're really solving a whole bunch of

problems at once by leaving omega explicitly as a variable.

Okay, now, hopefully that didn't completely lose you in in the weeds, but

let's just proceed, and I think sometimes it's better to just kind of plow through

and start using things and some of the more subtle points will become more clear

as you become more familiar with what's going on.

So the first thing I'm going to do this I'm going to drop the omega dependence

explicitly and just use I with a tilde and V with a tilde to represent the

phasor current and voltage. Now let's go back and look at a capacitor

again And we had the expression that related the current and voltage for a

capacitor. It's I is C times the time rate of change

of the voltage. Now, the nice thing here is if I assume

this form, with an E to the J omega T and assume that the I and V in front, this I

and this V. Are not time dependent.

They're just functions of frequency. I can take that derivative.

So Ie to the j omega t. That's the I.

Is C times ddt. The time derivative of V.