Question 7.19: Consider the circuit in Fig. 7.73, and assume that R1 = 1.5M...

(a) The smallest value for $R_{2}$ is 0 Ω, and the corresponding time constant for the circuit is

The largest value for $R_{2}$ is 2.5 MΩ, and the corresponding time constant for the circuit is

Thus, by proper circuit design, the time constant can be adjusted to introduce a proper time delay in the circuit.
(b) Assuming that the capacitor is initially uncharged, $v_{C}(0)=0$, while $v_{C}(\infty)=110$. But

where τ = 0.4 s, as calculated in part (a). The lamp glows when $v_{C}=70 V$ . If $v_{C}(t)=70 V \text { at } t=t_{0}$, then

or

Taking the natural logarithm of both sides gives

A more general formula for finding $t_{0}$ is

The lamp will fire repeatedly every τ seconds if and only if $t_{0}<\tau$. In this example, that condition is not satisfied.