Question

(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.

Short answer

1. In fig $27-18a,$the resistances $R_1\&R_3$are not is series
2. In fig,$27-18a$ the resistances$R_1\&R_2$are in parallel
3. The equivalent resistance of all four circuits is same

Step-by-step solution

Step 1: Given

Fig. 27-18

Determining the concept

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:

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

(a) Determining in fig 27-18a , are resistances  R1& R3 in series

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

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

(b) Determining In fig 27-18a, are resistances  R1& R2 in parallel

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.