Chapter 19, Problem 30P

To determine

The magnitudes and direction of current across each resistor in the given circuit if each batter has internal resistance of 1 ohm.

Answer to Problem 30P

The magnitudes and direction of current across each resistor in the given circuit are $I_1=-0.1316\text{A}$ , $I_2=-0.3099\text{A}$ , and $I_3=-0.1783\text{A}$ .

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.

  $\begin{array}{c}-25I_1-18I_2-1\cdot I_1+9=0\\ 26I_1+18I_2=9\end{array}$    … (1)

Apply Kirchhoff’s voltage law in the lower loop.

  $\begin{array}{c}18I_2+35I_3+1\cdot I_3-12=0\\ 18I_2+36I_3=12\end{array}$    … (2)

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

  $I_2=I_1+I_3$    … (3)

Substitute the value from equation (3) to equation (1).

  $\begin{array}{c}18I_2+36\left(I_2-I_1\right)=12\\ 18I_2+36I_2-36I_1=12\\ -36I_1+54I_2=12\end{array}$    … (4)

On solving equation (1) and (4),

  $\begin{array}{l}{I}_1=-0.1316\text{A}\\ I_2=-0.3099\text{A}\end{array}$

Substitute the values in equation (3).

  $\begin{array}{c}-0.3099=-0.1316+I_3\\ I_3=-0.1783\text{A}\end{array}$

Therefore, the magnitudes and direction of current across each resistor in the given circuit are $I_1=-0.1316\text{A}$ , $I_2=-0.3099\text{A}$ , and $I_3=-0.1783\text{A}$ .