Problem $2$: AC Power Analysis

$[9$ pts$]$

$($a$)[1$ pt$]$ Consider a resistor R driven by a sinusoidal voltage source V $($t$)=$ V$0$ sin$(2$f t$),$ where V$0$ is the

amplitude of the sinewave and f is the frequency. Write an expression for the instantaneous power

dissipated in the resistor as a function V$0,$ R$,$ f $,$ and t$.$

$($b$)[3$ pts$]$ Using the identity sin$2($x$)=1$cos$(2$x$)$

$2$ and the expression you found above, find the average

power dissipated in the resistor in terms of V$0$ and R$.$

$($c$)[1$ pt$]$ What voltage is needed to dissipate $1$W across a $100$ resistor?

$($d$)[2$ pts$]$ The Root Mean Square $($RMS$)$ voltage is the effective voltage that delivers the same power as

a DC voltage. For a sinusoidal voltage, find the RMS voltage as a function of the amplitude, V$0.$

$($e$)[2$ pts$]$ In the US$,$ the voltage in the wall is Vrms $=120$V $.$ What is the amplitude of the voltage? What

is the peak$-$to$-$peak $($highest positive to highest negative$)$ voltage? Why do you think the RMS value

is used to describe the voltage instead of the amplitude?

$4$