Problem

In the circuit shown, find the current flowing through the resistors.

Data

• EMF: $\varepsilon_1 = 10 \, \mathrm{V}$, $\varepsilon_2 = 6 \, \mathrm{V}$
• Resistances: $R_1 = 60 \, \Omega$, $R_2 = 10 \, \Omega$

Prerequisite Concepts

1. Apply Kirchhoff’s Voltage Law (KVL) in both loops to determine currents $I_1$ and $I_2$.

Solution

Loop 1 (abeda):

$$-\varepsilon_1 - (I_1 - I_2) R_1 = 0 \implies -10 - (I_1 - I_2)(60) = 0 \tag{1}$$

Loop 2 (befcb):

$$-I_2 R_2 + \varepsilon_2 - (I_1 - I_2) R_1 = 0 \implies -I_2 (10) + 6 - (I_1 - I_2)(60) = 0 \tag{2}$$

From equations (1) and (2), solve for $I_1$ and $I_2$:

• $I_2 = -4 \, \mathrm{A}$ (current through $R_2$)
• $I_1 = -9 \, \mathrm{A}$ (current through $R_1$).

Answer

• Current through $R_2$: $I_2 = -4 \, \mathrm{A}$
• Current through $R_1$: $I_1 = -9 \, \mathrm{A}$

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