(II) Determine the magnitudes and directions of the currents...

(II) Determine the magnitudes and directions of the currents in each resistor shown in Fig. $26-52 .$ The batteries have emfs of $\mathscr{E}_{1}=9.0 \mathrm{~V} \quad$ and $\quad \mathscr{E}_{2}=12.0 \mathrm{~V}$ and the resistors have values of $R_{1}=25 \Omega$, $R_{2}=48 \Omega, \quad$ and $\quad R_{3}=35 \Omega$ (a) Ignore internal resistance of the batteries. (b) Assume each battery has internal resistance $r=1.0 \Omega$.

(II) Determine the magnitudes and directions of the currents in each resistor shown in Fig. $26-52 .$ The batteries have emfs of $\mathscr{E}_{1}=9.0 \mathrm{~V} \quad$ and $\quad \mathscr{E}_{2}=12.0 \mathrm{~V}$ and the resistors have values of $R_{1}=25 \Omega$, $R_{2}=48 \Omega, \quad$ and $\quad R_{3}=35 \Omega$
(a) Ignore internal resistance of the batteries. (b) Assume each battery has internal resistance $r=1.0 \Omega$.

Key Concepts

Kirchhoff’s Circuit Laws

These are fundamental rules used to analyze circuits. Kirchhoff’s Voltage Law (KVL) states that the sum of all the voltage drops in a closed loop must equal the sum of the electromotive forces (emfs) in that loop, ensuring energy conservation. Kirchhoff’s Current Law (KCL) states that the total current entering any junction must equal the total current leaving that junction. Together, these laws allow one to set up equations for the unknown currents and voltages in a complex circuit.

Electromotive Force (EMF)

Electromotive force represents the voltage provided by a battery or other power source. It is the driving factor that pushes current through the circuit. When analyzing circuits with multiple EMFs, understanding how their magnitudes and polarities interact is crucial for determining the net voltage driving the currents in different branches of the circuit.

Internal Resistance

Internal resistance refers to the inherent resistance within a battery or power source that causes a voltage drop inside the source itself when current flows. This concept is important for accurately modeling real-world batteries, as the effective voltage supplied to the circuit is reduced by the drop across the internal resistance, which can significantly affect current distribution in the circuit.

Ohm’s Law

Ohm’s Law is a basic principle used in electrical circuit analysis, stating that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, with the resistance (R) being the constant of proportionality (V = IR). This law is essential for calculating the current through each resistor once the voltage drops across them are known.

Mesh/Loop Analysis

Mesh or loop analysis is a systematic method for analyzing circuits by applying Kirchhoff’s Voltage Law around independent loops in the circuit. By setting up and solving a system of equations based on these loops, it becomes possible to determine the currents in various branches, even in complex circuits with multiple batteries and resistors.

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