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Kirchhoff's Voltage Law

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Question

Find the potential difference $V_{a} - V_{b}$ in the circuits shown if figure (32-E12).

Figure 32 - E12

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Solution

In circuit ab baa. $E_{2} + i R_{2} + i R_{3} = 0$

$i R_{2} + i_{1} R_{3} = - E_{2}$

$\Rightarrow i R_{2} R_{1} + i R_{3} R_{1} = - E_{2} R_{1} . . . (1)$

$and i_{1} R_{3} + E_{1} - (i - i_{1}) R_{1} = 0$

$E_{1} = i_{1} (R_{1} + R_{3}) - i R_{1}$

$\Rightarrow E_{1} R_{2} = i R i R_{2} + i_{1} R_{2} R_{3} - i R_{1} R_{2} . . . (2)$

Adding (1) and (2), we get

$i_{1} R_{1} R_{2} + i_{1} R_{3} R_{1} + i_{1} R_{2} R_{3}$

$= E_{1} R_{2} - E_{2} R_{1}$

$\Rightarrow i_{1} (R_{3} R_{1} + R_{2} R_{1} + R_{2} R_{3})$

$= E_{1} R_{2} - E_{2} R_{1}$

$i_{1} = \frac{E_{1} R_{2} - E_{2} R_{1}}{R_{3} R_{1} + R_{2} R_{1}} + R_{2} R_{3}$

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