Chapter 17, Problem 45E

(a)

To determine

The current through the $2\text{ohm}$ resistor.

Answer to Problem 45E

The current through the $2\text{ohm}$ resistor is $\underset{\_}{1.5\text{A}}$.

Explanation of Solution

The circuit is given in Figure 17.28.

Let us represent the $6\text{ohm}$ resistor as $R_1$, the $3\text{ohm}$ resistor as $R_2$, and the $2\text{ohm}$ resistor as $R_3$.

$R_1$ and $R_2$ are connected parallel and $R_3$ is in series to the combination of $R_1$ and $R_2$. Thus the equivalent resistance of the circuit can be obtained as,

  $R_{\text{eq}}=R_{1,2}+R_3$        (I)

Here, $R_{\text{eq}}$ is the equivalent resistance, and $R_{1,2}$ is the total resistance of the combination of $R_1$ and $R_2$.

Write the expression for total resistance of the combination of $R_1$ and $R_2$ which are connected in parallel.

  $R_{1,2}=\frac{R_1{R}_2}{R_1+R_2}$        (II)

Use equation (II) in (I).

  $R_{\text{eq}}=\frac{R_1{R}_2}{R_1+R_2}+R_3$        (III)

Since $R_3$ is in series with the combination of $R_1$ and $R_2$, the current through $R_3$ and the current through the combination of $R_1$ and $R_2$ will be same and is given by,

  $I=\frac{V}{R_{\text{eq}}}$        (IV)

Here, $I$ is the current, and $V$ is the voltage difference across the whole circuit.

Conclusion:

Substitute $6\text{ohm}$ for $R_1$, $3\text{ohm}$ for $R_2$, and $2\text{ohm}$ for $R_3$ in equation (III) to find $R_{\text{eq}}$.

  $\begin{array}{c}{R}_{\text{eq}}=\frac{\left(6\text{ohm}\right)\left(3\text{ohm}\right)}{\left(6\text{ohm}\right)+\left(3\text{ohm}\right)}+\left(2\text{ohm}\right)\\ =4\text{ohm}\end{array}$

Substitute $4\text{ohm}$ for $R_{\text{eq}}$, $6\text{V}$ for $V$ in equation (IV) to find the current through the $2\text{ohm}$ resistor $I$.

  $\begin{array}{c}I=\frac{6\text{V}}{4\text{ohm}}\\ =1.5\text{A}\end{array}$

Therefore, the current through the $2\text{ohm}$ resistor is $\underset{\_}{1.5\text{A}}$.

(b)

To determine

The voltage difference across the $3\text{ohm}$ resistor.

Answer to Problem 45E

The voltage difference across the $3\text{ohm}$ resistor is $\underset{\_}{3\text{V}}$.

Explanation of Solution

The current through the combination is $1.5\text{A}$ since it is same as that through $R_3$. Since $R_1$ and $R_2$ are connected in parallel, the voltage difference across both of them will be same and is given by,

  $V_{1,2}=I{R}_{1,2}$        (V)

Here, $V_{1,2}$ is the voltage across the combination of $R_1$ and $R_2$.

Use equation (II) in (V) and express $V_{1,2}$ as $V_2$ since the voltage difference is same for $R_1$ and $R_2$.

  $V_2=I\left(\frac{R_1{R}_2}{R_1+R_2}\right)$        (VI)

Conclusion:

Substitute $6\text{ohm}$ for $R_1$, $3\text{ohm}$ for $R_2$, and $1.5\text{A}$ for $I$ in equation (VI) to find $V_2$.

  $\begin{array}{c}{V}_2=\left(1.5\text{A}\right)\left(\frac{\left(6\text{ohm}\right)\left(3\text{ohm}\right)}{\left(6\text{ohm}\right)+\left(3\text{ohm}\right)}\right)\\ =3\text{V}\end{array}$

Therefore, the voltage difference across the $3\text{ohm}$ resistor is $\underset{\_}{3\text{V}}$.

(c)

To determine

The current through the $3\text{ohm}$ resistor.

Answer to Problem 45E

The current through the $3\text{ohm}$ resistor is $\underset{\_}{1\text{A}}$.

Explanation of Solution

It is obtained that the voltage across the $3\text{ohm}$ resistor is $3\text{V}$.

Write the expression for the current through $R_2$, the $3\text{ohm}$ resistor.

  $I_2=\frac{V_2}{R_2}$        (VII)

Conclusion:

Substitute $3\text{V}$ for $V_2$, and $3\text{ohm}$ for $R_2$ in equation (VII) to find $I_2$.

  $\begin{array}{c}{I}_2=\frac{3\text{V}}{3\text{ohm}}\\ =1\text{A}\end{array}$

Therefore, the current through the $3\text{ohm}$ resistor is $\underset{\_}{1\text{A}}$.