A resistor with resistance R and a capacitor with capacitanc

A resistor with resistance R and a capacitor with capacitance C are connected in series to an AC voltage source. The time-dependent voltage across the capacitor is given by VC(t)=VC0sin?t.

Part C

Solution

Vc() = Vco
with the voltage across the capacitor determining our phase reference - not that we actually need to keep explicit track of the phase; we are asked for amplitudes only. In any case, the current through the capacitor will be given by
I() = Vc()/Zc()
but for a capacitor,
Zc() = jXc()
where Zc() is the (complex) impedance of the capacitor at angular frequency ,
j = -1 (Electrical engineers prefer using j for this constant rather than i because the convention for i being a variable name for current had already been established before complex arithmetic started being applied to the problem.)
and
Xc() is the (negative real) reactance of an ideal capacitor at angular frequency , which is -1/C.

The current through the capacitor, wich is the same as the current through the circuit as a whole because we are talking about a series circuit here, is therefore
I() = Vco / (-j/C)
= jCVco

The voltage across the resistor will then be
Vr() = I()Zr()
= I()R
= jRCVco
and
|Vr()| = RCVco
If you want the voltage as a function of time, it will be
Vr(t) = RCVco cos (t) or
RCVco sin (t + /2)

or V_{R =} V_coCR

V_R=0.1*100*10^-12*3000*10^5

= 3*10^-3