Chapter 19, Problem 25P

(a)

To determine

The potential difference between points a and d.

Answer to Problem 25P

The potential difference between points a and d is $25.8\text{V}$ .

Explanation of Solution

Given information:

Consider the given circuit.

Formula used:

Apply Kirchhoff’s voltage law in the circuit to determine the currents.

Calculation:

Apply Kirchhoff’s voltage law in the upper loop.

  $\begin{array}{c}-45-1\cdot I_3-40I_3-30I{1}_{}=0\\ 30I_1+41I_3=45\end{array}$  … (1)

Apply Kirchhoff’s voltage law in the lower loop.

  $\begin{array}{c}-40I_3-1\cdot I_3-45-20I_2-1\cdot I_2-80=0\\ 21I_2+41I_3=125\end{array}$   … (2)

At node (a), apply Kirchhoff’s current law.

  $I_3=I_1+I_2$  … (3)

Substitute the value of $I_3$ from equation (3) to equation (1) and (2).

  $\begin{array}{c}30I_1+41\left(I_1+I_2\right)=45\\ 71I_1+41I_2=45\end{array}$

and,

  $\begin{array}{c}21I_2+41\left(I_1+I_2\right)=125\\ 21I_2+41I_1+41I_2=125\\ 41I_1+62I_2=125\end{array}$

On solving the above equations,

  $\begin{array}{c}{I}_1=0.86\text{A}\\ I_2=-2.58\text{A}\end{array}$

Substitute the values in equation (3).

  $\begin{array}{c}{I}_3=0.86-2.58\\ =-1.72\text{A}\end{array}$

Calculate the voltage drop across the points a and d.

  $V_{ad}=30I_1$

Substitute the value in the above expression.

  $\begin{array}{c}{V}_{ad}=30\left(0.86\right)\\ =25.8\text{V}\end{array}$

Therefore, thepotential difference between points a and d is $25.8\text{V}$ .

(b)

To determine

The terminal voltage of each battery.

Answer to Problem 25P

The terminal voltage of the 45-volt battery is $77.42\text{V}$ and the 80-volt battery is $77.42\text{V}$ .

Explanation of Solution

Given information:

Consider the given circuit.

Formula used:

Apply Kirchhoff’s voltage law in the circuit to determine the terminal voltage across each battery.

Calculation:

From part (a), the values of current are as follows. On solving the above equations,

  $\begin{array}{c}{I}_1=0.86\text{A}\\ I_2=-2.58\text{A}\\ I_3=-1.72\text{A}\end{array}$

Calculate the terminal voltage of 80-volt battery.

  $\begin{array}{c}{V}_{80}=80-1\cdot \left(2.58\right)\\ =77.42\text{V}\end{array}$

Similarly, calculate the terminal voltage of 45-volt battery.

  $\begin{array}{c}{V}_{45}=45-1\cdot \left(1.72\right)\\ =43.28\text{V}\end{array}$

Therefore, the terminal voltage of the 45-volt battery is $77.42\text{V}$ and the 80-volt battery is $77.42\text{V}$ .