Chapter 8.2, Problem 20P

Consider the following special cases of the simple series circuit [Figure 1.1 and equation $(1.2)].$

(a) $RC$ circuit (that is, $L=0$ ) with $V=0;$ find $q$ as a function of $t$ if $q_0$ is the charge on the capacitor at $t=0.$

(b) $RL$ circuit (that is, no capacitor; this means $1/C=0$ ) with $V=0;$ find $I(t)$ given $I=I_0$ at $t=0$.

(c) Again note that these are the same differential equations as in Problem 19 and Example 1. The terminology is again different; we define the time constant $\tau$ for a circuit as the time required for the charge (or current) to fall to $1/e$ times its initial value. Find the time constant for the circuits (a) and (b). If the same equation, say $y=y_0{e}^{-at},$ represented either radioactive decay or light absorption or an RC or RL circuit, what would be the relations among the half-life, the half-value thickness, and the time constant?