How much current flows through the bottom wire in FIGURE P28.66, and in which direction?

Verified step by step guidance

1. Step 1: Analyze the circuit configuration. The circuit consists of two voltage sources (9 V and 15 V) and resistors arranged in a combination of series and parallel connections. The bottom wire connects the 12 Ω and 24 Ω resistors in the triangular section of the circuit.
2. Step 2: Apply Kirchhoff's Voltage Law (KVL) to the loops in the circuit. KVL states that the sum of the potential differences (voltage drops and rises) around any closed loop is zero. Identify the two main loops: one containing the 9 V battery and the 6 Ω resistor, and the other containing the 15 V battery and the 10 Ω resistor.
3. Step 3: Calculate the equivalent resistance of the triangular section. The 12 Ω and 24 Ω resistors are connected in parallel. Use the formula for parallel resistance: $R_{\mathrm{eq}}=\frac{1}{\frac{1}{12}+\frac{1}{24}}$. Simplify to find the equivalent resistance.
4. Step 4: Determine the current in each loop using Ohm's Law ($I=\frac{V}{R}$). For each loop, calculate the total resistance and use the voltage of the respective battery to find the current.
5. Step 5: Use Kirchhoff's Current Law (KCL) at the junctions to find the current through the bottom wire. KCL states that the sum of currents entering a junction equals the sum of currents leaving the junction. Combine the currents from the two loops and determine the direction of the current through the bottom wire based on the net flow.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as I = V/R. Understanding this law is crucial for calculating the current in the circuit based on the voltage sources and resistances present.

Series and Parallel Circuits

In electrical circuits, components can be arranged in series or parallel configurations. In a series circuit, the current is the same through all components, while the total resistance is the sum of individual resistances. In a parallel circuit, the voltage across each component is the same, and the total current is the sum of the currents through each branch. Recognizing the arrangement of resistors in the given circuit is essential for determining the current through the bottom wire.

Kirchhoff's Laws

Kirchhoff's Laws consist of two principles that govern the conservation of charge and energy in electrical circuits. Kirchhoff's Current Law (KCL) states that the total current entering a junction must equal the total current leaving it, while Kirchhoff's Voltage Law (KVL) states that the sum of the electrical potential differences around any closed circuit loop must equal zero. These laws are fundamental for analyzing complex circuits and finding unknown currents and voltages.