A `200 Omega` resistor is connected in series with a `5(mu)F` capacitor. The voltage across the resistor is `V_(R )=(1.20 V) cos (2500 rad s^(-1))t`.
(a) Derive an expression for the circuit current.
(b) Determine the capactive reactance of the capacitor.
(c ) derive an experssion for the voltage across the capacitor.

Explanation

This is a series curcuit. The current is the same thrugh the capacitor as through the resistor. Our target variables are current I, capactive reactance `X_(C)` and capacitor voltage `V_(C)`.

(a) using `V_(R )-iR,` we find that current i in the resistor and through the circuit as a whole is
`i-(V_R)/(R )=((1.20 V)cos(2500 rad s^(-1))t)/(200 Omega)`
`=(6.0xx10^(-3)A) cos (2500 rad s^(-1))t`
(b) The capacitive reactance at `omega=2500 rad s^(-1)` is
`(X_C)=(1)/(omega C)=(1)/((2500 rad s^(-1)_(5.0xx10^(-6)F))=80 Omega`
(c ) The amplitude `V_(C)` of hte voltage across the capacitor is
`V_(C0)=I_(0)X_(C)=(6.0xx10^(-3)A)(80 Omega)=0.48V`
The instantaneous capcitor voltage `V_(C)` is given by
`V_(C)=V_(C0) cos(omega t-90^(@))`
`=(0.48 V)cos[(25oo rad s^(-1))t-(pi)//(2)rad]`.