Chapter Questions
Judging on the basis of your experimental results, under what conditions are the charging times of different $R C$ circuits the same?
Show that the magnitude of the charge on a capacitor is given by $Q=Q_{0}\left(1-e^{-\mathrm{s} / \tau}\right)$ and $Q=Q_{0} e^{-t / \tau}$ for charging and discharging, respectively.
In the form $V=V_{0}\left(1-e^{-t / \tau}\right)$, the $\tau=R C$ in the exponential must have units of time. (Why?) Show that this is the case.
What is the voltage across a capacitor after a time of two constants when (a) charging from zero voltage and (b) discharging from a fully charged condition?
How could the value of an unknown capacitance be determined using the experimental procedures? Show explicitly by assuming a value for an experimentally determined time constant.
With $V=V_{o} e^{-t / R C}$, it mathematically takes an infinite time for a capacitor in an $R C$ circuit to discharge. Practically, how many time constants does it take for a capacitor to discharge to less than $1 \%$ of its initial voltage?
$$\begin{array}{l}\text { Show that the time for the voltage in the } R C \text { circuit to rise to } V_{\mathrm{o}} / 2 \text { ("half-max") }\\\text { is } t_{1 / 2}=\tau \ln 2\end{array}$$
A $2.0-\mu \mathrm{F}$ capacitor in a circuit in series with a resistance of $1.0 \mathrm{M} \Omega$ is charged with a $6.0$ - $\mathrm{V}$ battery. How long would it take to charge the capacitor to three-fourths of its maximum voltage?