Chapter 17, Problem 32E

(a)

To determine

The EMF $\epsilon$ in the given circuit.

Answer to Problem 32E

The EMF $\epsilon$ in the given circuit is $\underset{\_}{-6\text{V}}$.

Explanation of Solution

The circuit is shown in Figure 17.25. Let us represent the $1\text{ohm}$ resistance as $R_1$, $3\text{ohm}$ resistance as $R_2$, $2\text{V}$ battery as $V_2$, and the current as $I$.

The current is flowing in the anti-clockwise direction in the loop. Apply Kirchhoff’s voltage law to the given circuit.

  $V_2-I{R}_2-I{R}_1-\epsilon =0$        (I)

Solve equation (I) for $\epsilon$.

  $\epsilon =V_2-I{R}_1-I{R}_2$        (II)

Conclusion:

Substitute $2\text{V}$ for $V_2$, $2\text{A}$ for $I$, $1\text{ohm}$ for $R_1$, and $3\text{ohm}$ for $R_2$ in equation (II) to find $\epsilon$.

  $\begin{array}{c}\epsilon =2\text{V}-\left(2\text{A}\right)\left(1\text{ohm}\right)-\left(2\text{A}\right)\left(3\text{ohm}\right)\\ =-6\text{V}\\ \left|\epsilon \right|=6\text{V}\end{array}$

Therefore, the EMF $\epsilon$ in the given circuit is $\underset{\_}{6\text{V}}$.

(b)

To determine

The power supplied to each of the two resistors.

Answer to Problem 32E

The power supplied to the $1\text{ohm}$ resistor is $\underset{\_}{4\text{W}}$, and that to the $3\text{ohm}$ resistor is $\underset{\_}{12\text{W}}$

Explanation of Solution

Given that the current through each of the resistors is $2\text{A}$.

Write the expression for the power supplied to a resistor.

  $P=I^{2}R$        (III)

Here, $P$ is the power, $I$ is the current, and $R$ is the resistance.

The power supplied to each the resistors can be computed from equation (III) as,

  $\begin{array}{l}{P}_1=I^2{R}_1\\ P_2=I^2{R}_2\end{array}\}$        (IV)

Conclusion:

Substitute $2\text{A}$ for $I$, $1\text{ohm}$ for $R_1$, and $3\text{ohm}$ for $R_2$ in equation (IV) to find the power supplied resistors.

  $\begin{array}{c}{P}_1={\left(2\text{A}\right)}^2\left(1\text{ohm}\right)\\ =4\text{W}\end{array}$

  $\begin{array}{c}{P}_2={\left(2\text{A}\right)}^2\left(3\text{ohm}\right)\\ =12\text{W}\end{array}$

Therefore, the power supplied to the $1\text{ohm}$ resistor is $\underset{\_}{4\text{W}}$, and that to the $3\text{ohm}$ resistor is $\underset{\_}{12\text{W}}$

(c)

To determine

The power supplied by each EMFs in the circuit and compare these with the power suppled to each resistors.

Answer to Problem 32E

The power supplied by the $6\text{V}$ battery is $\underset{\_}{12\text{W}}$, and that by the $2\text{V}$ battery is $\underset{\_}{4\text{W}}$. The power supplied to the $1\text{ohm}$ resistor is same as the power supplied by the $2\text{V}$ EMF. The power supplied to the $3\text{ohm}$ resistor is same as the power supplied by the $6\text{V}$ EMF.

Explanation of Solution

Write the expression for the power supplied by the battery (EMF).

  $P^{\prime }=IV$        (V)

Here, $P^{\prime }$ is the power supplied by the EMF,  $V$ is the potential difference (which is same as the EMF of the battery)

The power supplied to each battery can be computed from equation (V) as,

  $\begin{array}{l}{P}_1{}^{\prime }=I{V}_1\\ P_2{}^{\prime }=I{V}_2\end{array}\}$        (VI)

Conclusion:

Substitute $2\text{A}$ for $I$, $6\text{V}$ for $V_1$, and $2\text{V}$ for $V_2$ in equation (VI) to find the power supplied EMFs.

  $\begin{array}{c}{P}_1{}^{\prime }=\left(2\text{A}\right)\left(6\text{V}\right)\\ =12\text{W}\end{array}$

  $\begin{array}{c}{P}_2{}^{\prime }=\left(2\text{A}\right)\left(2\text{V}\right)\\ =4\text{W}\end{array}$

The power supplied to the $1\text{ohm}$ resistor is same as the power supplied by the $2\text{V}$ EMF. The power supplied to the $3\text{ohm}$ resistor is same as the power supplied by the $6\text{V}$ EMF.

Therefore, The power supplied by the $6\text{V}$ battery is $\underset{\_}{12\text{W}}$, and that by the $2\text{V}$ battery is $\underset{\_}{4\text{W}}$. The power supplied to the $1\text{ohm}$ resistor is same as the power supplied by the $2\text{V}$ EMF. The power supplied to the $3\text{ohm}$ resistor is same as the power supplied by the $6\text{V}$ EMF.