In the circuit shown in Fig. E26.27
find (b) the unknown emfs ε_1 and ε_2.

Verified step by step guidance

1. Identify the currents and resistances in the circuit. The currents are given as 1.75 A and 1.25 A, and the resistances are 8.0 Ω, 1.0 Ω, and 8.0 Ω.
2. Apply Kirchhoff's Voltage Law (KVL) to the left loop: E1 - (1.75 A * 8.0 Ω) - (1.75 A * 8.0 Ω) = 0.
3. Simplify the equation from step 2 to find E1.
4. Apply Kirchhoff's Voltage Law (KVL) to the right loop: E2 - (1.25 A * 1.0 Ω) - (1.25 A * 8.0 Ω) = 0.
5. Simplify the equation from step 4 to find E2.

Key Concepts

Kirchhoff's Loop Rule

Kirchhoff's Loop Rule states that the sum of the potential differences (voltage) around any closed loop in a circuit must equal zero. This principle is based on the conservation of energy, indicating that the energy supplied by the sources (like batteries) is equal to the energy consumed by the resistors in the loop.

Ohm's Law

Ohm's Law relates voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. This fundamental relationship allows us to calculate the voltage across a resistor when the current flowing through it and its resistance are known, which is essential for analyzing circuits.

Series and Parallel Resistors

In circuits, resistors can be arranged in series or parallel configurations. In a series circuit, the total resistance is the sum of individual resistances, while in a parallel circuit, the total resistance can be found using the formula 1/R_total = 1/R1 + 1/R2 + ... This distinction is crucial for calculating the overall resistance and current in the circuit.