Chapter 2.4, Problem 22EQ

The networks in parts (a) and (b) of Figure 2.23 show two resistors coupled in series and in parallel, respectively. We wish to find a general formula for the effective resistance of each network—that is, find $R_{\text{eff}}$ such that $E=R_{\text{eff}}I$.

1. Show that the effective resistance $R_{\text{eff}}$ of a network with two resistors coupled in series [Figure 2.23(a)] is given by

   $R_{\text{eff}}=R_1+R_2$

2. Show that the effective resistance $R_{\text{eff}}$ of a network with two resistors coupled in parallel [Figure 2.23(b)] is given by

   ${\mathrm{R}}_{\text{eff}}=\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}$