Problem 49 - Physics

Question

A \(16-\Omega\) loudspeaker and an \(8.00-\Omega\) loudspeaker are connected in parallel across the terminals of an amplifier. Assuming the speakers behave as resistors, determine the equivalent resistance of the two speakers.

Step-by-Step Solution

Answer

The equivalent resistance is approximately 5.33 ohms.

Step 1: Identify the Formula for Parallel Resistance

To find the equivalent resistance of resistors in parallel, we use the formula: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \), where \( R_{eq} \) is the equivalent resistance, and \( R_1 \) and \( R_2 \) are the resistances of the loudspeakers.

Step 2: Substitute the Given Values

Substitute the values given in the problem into the formula: \( R_1 = 16 \) ohms and \( R_2 = 8 \) ohms. The equation becomes: \( \frac{1}{R_{eq}} = \frac{1}{16} + \frac{1}{8} \).

Step 3: Calculate Each Fraction

Calculate the reciprocal of each resistance: \( \frac{1}{16} = 0.0625 \) and \( \frac{1}{8} = 0.125 \).

Step 4: Add the Fractions

Add the two fractions together: \( 0.0625 + 0.125 = 0.1875 \).

Step 5: Find the Reciprocal to Determine the Equivalent Resistance

The equivalent resistance \( R_{eq} \) is the reciprocal of the sum: \( R_{eq} = \frac{1}{0.1875} \). Calculate \( \frac{1}{0.1875} \) to get \( R_{eq} = 5.333 \) ohms.

Key Concepts

Understanding a Parallel Circuit

In electrical engineering, a parallel circuit is one where multiple components are connected. Each of these components shares the same two terminals.
This means the voltage across each component is the same. However, the total current flowing through the circuit is divided among the various branches.
This is different from a series circuit where components are connected one after the other, and the total current flows through each component without division.

• In a parallel configuration: if one component fails, the current can still pass through the others.
• Parallel circuits are commonly used in home wiring—ensuring appliances function independently.
• Understanding these circuits aids in finding the total resistance, or equivalent resistance.

This concept is key to understanding the behavior of electric currents in complex network configurations.

Resistance Calculation in Parallel Circuits

Calculating resistance in a parallel circuit might seem complicated, but the process is straightforward with a systematic approach. Unlike series circuits, where resistances are added directly, parallel circuits use reciprocal calculations to find the equivalent resistance.
The formula used is \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \). This formula represents the relationship between the total, or equivalent, resistance \( R_{eq} \) and the individual resistances \( R_1, R_2, \ldots, R_n \).

• For the exercise: \( R_1 = 16 \) ohms and \( R_2 = 8 \) ohms. Substitute these values into the formula.
• Calculate each fractional part as the reciprocal, meaning finding the inverse of each resistance.
• Add the results: \( \frac{1}{16} + \frac{1}{8} = 0.0625 + 0.125 = 0.1875 \).
• Finally, find the reciprocal of the sum to get the equivalent resistance: \( R_{eq} = \frac{1}{0.1875} = 5.333 \) ohms.

Practice with these calculations solidifies understanding and prepares you for more intricate circuit analysis.

Exploring Ohm's Law

Ohm's Law is a foundational principle in electromagnetism that relates voltage, current, and resistance in an electrical circuit. It is expressed as \( V = IR \), where \( V \) is the voltage across the circuit, \( I \) is the current flowing through it, and \( R \) is the resistance.
This law is essential because it provides a way to calculate an electric parameter when the other two are known.
In the context of parallel circuits, Ohm’s Law helps in understanding how the changes in resistance or voltage can affect the current in each branch of the circuit.

• If the equivalent resistance decreases, the current through the circuit increases for a constant voltage.
• Using Ohm's Law along with the resistance calculations enables precise design and analysis of circuits.
• By practicing these principles, you can predict circuit behavior under different conditions.

Whether you're working with simple or complex circuits, Ohm's Law remains an invaluable tool for electrical analysis and design.