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See the electrical circuit shown in the figure. Which of the following equations is correct for it?

1. \(\varepsilon_1 - (i_1 + i_2)R - i_1 r_1 = 0\)
2. \(\varepsilon_2 - i_2 r_2 - \varepsilon_1 - i_1 r_1 = 0\)
3. \(-\varepsilon_2 - (i_1 + i_2)R + i_2 r_2 = 0\)
4. \(\varepsilon_1 - (i_1 + i_2)R + i_1 r_1 = 0\)

Correct answer: \(\varepsilon_1 - (i_1 + i_2)R - i_1 r_1 = 0\)

Solution

Using Kirchhoff’s loop rule, the algebraic sum of emfs and potential drops in a closed loop must be zero. For the loop containing \(\varepsilon_1\), the external resistance \(R\), and internal resistance \(r_1\), the drops are \((i_1+i_2)R\) and \(i_1 r_1\). Therefore the correct equation is \(\varepsilon_1 - (i_1+i_2)R - i_1 r_1 = 0\).

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