(a) In Fig. 27-18a, are resistors$\mathbf{}{\mathbf{R}}_1$and ${\mathbf{R}}_3$in series?

(b) Are resistors $\mathbf{}{\mathbf{R}}_1\mathbf{\&}\mathbf{}{\mathbf{R}}_3$in parallel?

(c) Rank the equivalent resistances of the four circuits shown in Fig. 27-18, greatest first.

Fig. 27-18

Here, use the formula for the equivalent resistance for parallel as well as series arrangement.

For series arrangement
${\mathbf{R}}_{\mathrm{eq}}\mathbf{=}{\mathbf{R}}_1\mathbf{+}{\mathbf{R}}_2$

For parallel arrangement
$\frac{1}{{\mathbf{R}}_{\mathrm{eq}}}\mathbf{=}\frac{1}{{\mathbf{R}}_1}\mathbf{+}\frac{1}{{\mathbf{R}}_2}$

Formulae are as follow:

$$\begin{gathered} R_{eq}=R_1+R_2 \\ \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2} \end{gathered}$$

In $fig27-18a$, the resistances$R_1\&R_3$are not in series.

Hence, the resistances $R_1\&R_3$are not in series.

In $fig27-18a$, the resistances$R_1\&R_2$are in parallel.

Hence, theresistances $R_1\&R_2$are in parallel

Step 4: (b) Determining the rank of equivalent resistance of all the four circuits.

For circuit a:

$R_{1}and{R}_2$ are in parallel. So, the equivalent of this is R and as follow:

$$\begin{gathered} \frac{1}{R^{\text{'}}}=\frac{1}{R_1}+\frac{1}{R_2} \\ \frac{1}{R^{\text{'}}}=\frac{R_1+R_2}{R_1{R}_2} \\ {R}^{\text{'}}=\frac{R_1{R}_2}{R_1+R_2} \end{gathered}$$

Now, $R^{\text{'}}and{R}_3$are in the series. So, the equivalent of this R is as follow:

$$\begin{gathered} R=R_3+R\text{'} \\ R=R_3+\frac{R_1{R}_2}{R_1+R_2} \\ R=\frac{R_1{R}_3+R_3{R}_2+R_1{R}_2}{R_1+R_2}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \mathrm{..}\left(1\right) \end{gathered}$$

For circuit b:

$R_1\text{and}{R}_2$ are in parallel. So, the equivalent of this is $R\text{'}$and as follow:

$$\begin{gathered} \frac{1}{R^{\text{'}}}=\frac{1}{R_1}+\frac{1}{R_2} \\ \frac{1}{R^{\text{'}}}=\frac{R_1+R_2}{R_1{R}_2} \\ {R}^{\text{'}}=\frac{R_1{R}_2}{R_1+R_2} \end{gathered}$$

Now, $R^{\text{'}}and{R}_3$are in series. So, the equivalent of this R is as follow:

$$\begin{gathered} R=R_3+R\text{'} \\ R=R_3+\frac{R_1{R}_2}{R_1+R_2} \\ R=\frac{R_1{R}_3+R_3{R}_2+R_1{R}_2}{R_1+R_2}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(2\right) \end{gathered}$$

Fig c:

$R_{1}and{R}_2$ are in parallel. So, the equivalent of this is $R\text{'}$ and as follow:

$$\begin{gathered} \frac{1}{R^{\text{'}}}=\frac{1}{R_1}+\frac{1}{R_2} \\ \frac{1}{R^{\text{'}}}=\frac{R_1+R_2}{R_1{R}_2} \\ {R}^{\text{'}}=\frac{R_1{R}_2}{R_1+R_2} \end{gathered}$$

Now, $R^{\text{'}}and{R}_3$are in series. So, the equivalent of this R is as follow:

$$\begin{gathered} R=R_3+R\text{'} \\ R=R_3+\frac{R_1{R}_2}{R_1+R_2} \\ R=\frac{R_1{R}_3+R_3{R}_2+R_1{R}_2}{R_1+R_2}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .\left(3\right) \end{gathered}$$

Fig d:

$R_{1}and{R}_2$ are in parallel. So, the equivalent of this is $R\text{'}$and as follow:

$$\begin{gathered} \frac{1}{R^{\text{'}}}=\frac{1}{R_1}+\frac{1}{R_2} \\ \frac{1}{R^{\text{'}}}=\frac{R_1+R_2}{R_1{R}_2} \\ {R}^{\text{'}}=\frac{R_1{R}_2}{R_1+R_2} \end{gathered}$$

Now, $R^{\text{'}}and{R}_3$are in series. So, the equivalent of this R is as follow:

$$\begin{gathered} R=R_3+R\text{'} \\ R=R_3+\frac{R_1{R}_2}{R_1+R_2} \\ R=\frac{R_1{R}_3+R_3{R}_2+R_1{R}_2}{R_1+R_2}\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \mathrm{..}\left(4\right) \end{gathered}$$

Hence, from equations (1), (2), (3) and (4), we can conclude that the equivalent resistance of all four circuits is same.

Therefore, find equivalent resistance of all four arrangements and compare the results to rank them.