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

1. ε₂ − i₂ r₂ − ε₁ − i₁ r₁ = 0
2. ε₂ − (i₂ + i₁) R + i₂ r₂ = 0
3. ε₁ − (i₁ + i₂) R + i₁ r₁ = 0
4. ε₁ − (i₁ + i₂) R − i₁ r₁ = 0

Correct answer: ε₁ − (i₁ + i₂) R + i₁ r₁ = 0

Solution

Using Kirchhoff's Voltage Law (KVL), the sum of potential differences in a closed loop is zero. For the loop containing ε₁, (i₁ + i₂)R is the potential drop across the resistor, and i₁r₁ is the drop across the internal resistance. The correct equation is ε₁ − (i₁ + i₂)R + i₁r₁ = 0.

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