Chapter 13, Problem 4SP

In the combination of 12 Ω resistors shown in the diagram, there are two different parallel combinations that, in turn, are in series with the middle resistor.

a.    What is the resistance of each of the two parallel combinations?

b.    What is the total equivalent resistance between points A and B?

c.    If there is a voltage difference of 18 V between points A and B, what is the current flowing through the entire combination?

d.    What is the current flowing through each of the resistors in the three-resistor parallel combination?

To determine

The equivalent resistance of the two parallel segments.

Answer to Problem 4SP

The current through the circuit is $0.057\text{A}$.

Explanation of Solution

Given info: The given circuit is shown below.

Write the formula for the resultant resistance of two equal resistors connected in parallel.

$R_2=\frac{R}{2}$

Here,

$R_2$ is the equivalent resistance of the two equal parallel resistors

$R$ is the resistance of each resistor

Substitute $12\text{Ω}$ for $R$ to determine the equivalent resistance of the parallel connection of two $12\text{Ω}$.

$\begin{array}{c}{R}_2=\frac{12\text{Ω}}{3}\\ =6\text{Ω}\end{array}$

Write the formula for the resultant resistance of three equal resistors connected in parallel.

$R_3=\frac{R}{3}$

Here,

$R_3$ is the equivalent resistance of the three equal parallel resistors

$R$ is the resistance of each resistor

Substitute $12\text{Ω}$ for $R$ to determine the equivalent resistance of the parallel cpnnection of three $12\text{Ω}$.

$\begin{array}{c}{R}_3=\frac{12\text{Ω}}{3}\\ =4\text{Ω}\end{array}$

Conclusion:

The equivalent resistance of the two parallel connections in the arrangement is $6\text{Ω}$ and $4\text{Ω}$.

To determine

The equivalent resistance between point A to B.

Answer to Problem 4SP

The overall resistance between points A and B is $\text{22}\text{Ω}$.

Explanation of Solution

Given info: The given circuit is shown below.

In this arrangement the two parallel segments and one $12\text{Ω}$ are in series. The equivalent resistance will be the sum of the resistances of all the three segments from point A and B.

Write the formula for the equivalent resistance between point A and B.

$R=R_1+R_2+R_3$

Here,

$R$ is the equivalent resistance between point A and B

$R_1$ is the resistance of the first segment

$R_2$ is the resistance of the second segment

$R_3$ is the resistance of the third segment

Substitute $6\text{Ω}$ for $6\text{Ω}$ , $12\text{Ω}$ for $R_2$, $4\text{Ω}$ for $R_3$ to determine $R$.

$\begin{array}{c}R=6\text{Ω}+12\text{Ω}+4\text{Ω}\\ \text{=22}\text{Ω}\end{array}$

Conclusion:

The overall resistance between points A and B is $\text{22}\text{Ω}$.

To determine

The current flowing through the circuit.

Answer to Problem 4SP

The current in the circuit is $0.818\text{A}$.

Explanation of Solution

Given info: The voltage difference between point A and B is $18\text{V}$. The given circuit is shown below.

Write the formula for current.

$I=\frac{V}{R}$

Here,

$R$ is the equivalent resistance between point A and B

$I$ is the current

$V$ is the voltage difference

From section (b) the overall resistance between points A and B is $\text{22}\text{Ω}$.

Substitute $\text{22}\text{Ω}$ for $R$, $18\text{V}$ for $V$ to determine $I$.

$\begin{array}{c}I=\frac{18\text{V}}{\text{22}\text{Ω}}\\ =0.818\text{A}\end{array}$

Conclusion:

The current in the circuit is $0.818\text{A}$.

To determine

The current flowing through the individual resistors in the three-resistor parallel combination.

Answer to Problem 4SP

The current flowing through the individual resistors in the three-resistor parallel combination is $0.273\text{A}$.

Explanation of Solution

Given info: The given circuit is shown below.

The current through all the segments will be same. From section (c) the current in the circuit is $0.818\text{A}$. So the total current in the three-register parallel segment is $0.818\text{A}$.

The current through each segment of a parallel connection of resistors depends on the resistance of each resistor. Since the resistance of the three resistors is same, equal current will pass through the individual resistor which is one third of the total current.

Write the formula for the current through each of the resistor in the three-resistor parallel segment.

$I_i=\frac{I}{3}$

Here,

$I_i$ is the current through each resistor

$I$ is the total current

Substitute $0.818\text{A}$ for $I_i$ to determine $I$.

$\begin{array}{c}{I}_i=\frac{0.818\text{A}}{3}\\ =0.273\text{A}\end{array}$

Conclusion:

The current flowing through the individual resistors in the three-resistor parallel combination is $0.273\text{A}$.