The differential equation in the title of this thread is the usual starting point for the transformation of networks to the steady-state frequency domain. But if you had read further in the book it would have undoubtedly explained how to calculate the value of capacitive reactance, and how to apply Ohm's Law in the steady-state frequency domain with the calculated value of capacitive reactance, Xc, in ohms and constant RMS values for AC voltage and current at a single given frequency.
Somewhere in that book it will give the formula for capacitive reactance as Xc=1/(2πf⋅C)Ω or it may use ω in place of 2πf. Then just apply Ohm's Law. It really is just that simple.
If he could just learn to differentiate the voltage equation, then he would not have to get involved with reactance. By the way, Ohm's law is not R=V/I and all its variations, like most folks believe. That formula is the
definition of resistance or impedance. Ohm's law is a
property of a material, specifically its resistive linearity. A metal wire obeys Ohm's law, but a junction diode does not. I can show you quotes in a couple of good physics books that show that my statement to be true.
Also, regarding the flow of current through a capacitor: It is unquestionably true that for every electron that flows into one terminal of a capacitor, there is a corresponding electron that flows out of the other terminal. If you want to call that current flowing through the capacitor, then feel free to do so. Everyone else does, except for one known curmudgeon.
Anyone who says current exists through a capacitor is talking technical trash (TTT). A capacitor is not a frequency dependent resistor. A capacitor accumulates and depletes charge at the same time in a circuit. By doing so, it stores energy. If a capacitor allowed current to pass, it would instead be a resistor. Everyone should describe it correctly.
Ratch
