$(25\%)$ You have been given a capacitor circuit that has been designed to measure humidity. The

capacitance value varies as the humidity changes from $0$ to $100\%.$ The voltage across the capacitor

is measured in time after a step input of the source voltage Vs$.$ With this response data the time

constant of the system and the measured humidity can be determined.

RC dvC $($t$)$

dt $+$ vC $($t$)=$ Vs

R $=10$ k$,$ C $=10$ F

$($a$)$ Consider the rst order dierential equation above. What is the time constant, of the system?

$($b$)$ What is the static sensitivity of the system? What does this specic value mean?

$($c$)$ The source voltage and capacitor voltage are initially vC $($t $=0)=0\mathrm{.}0$ and Vs$($t $=0)=0\mathrm{.}0.$ At

time t $=0$ the source voltage of Vs $=7\mathrm{.}2$ is applied, and the capacitor voltage begins to increase.

Calculate the time t at which the capacitor voltage vC will reach $6\mathrm{.}5$ volts.

$2$