Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Electrical resistance In electrical circuits, the formula \(1 / R=\left(1 / R_{1}\right)+\left(1 / R_{2}\right)\) is used to find the total resistance \(R\) if two resistors \(R_{1}\) and \(R_{2}\) are connected in parallel. Given three resistors, A, B, and \(\mathrm{C}\), suppose that the total resistance is 48 ohms if A and B are connected in parallel, 80 ohms if \(\mathrm{B}\) and \(\mathrm{C}\) are connected in parallel, and \(60 \mathrm{ohms}\) if \(\mathrm{A}\) and \(\mathrm{C}\) are connected in parallel. Find the resistances of A, B, and C.

Step-by-Step Solution

Answer

The resistances are \(R_A = 120\), \(R_B = 80\), \(R_C = 240\) ohms.

Step 1: Write down the formulas

We have three scenarios: A and B in parallel, B and C in parallel, and A and C in parallel. For each of these scenarios, we can use:1. \(\frac{1}{R} = \frac{1}{R_A} + \frac{1}{R_B}\) for A and B yielding \(R = 48\).2. \(\frac{1}{R} = \frac{1}{R_B} + \frac{1}{R_C}\) for B and C yielding \(R = 80\).3. \(\frac{1}{R} = \frac{1}{R_A} + \frac{1}{R_C}\) for A and C yielding \(R = 60\).

Step 2: Express the formulas for each pair

Convert each formula into an equation:1. \(\frac{1}{48} = \frac{1}{R_A} + \frac{1}{R_B}\)2. \(\frac{1}{80} = \frac{1}{R_B} + \frac{1}{R_C}\)3. \(\frac{1}{60} = \frac{1}{R_A} + \frac{1}{R_C}\)

Step 3: Simplify the equations

Write each equation in terms of known values, getting common denominators:1. Solve the equation for A and B: \(\frac{R_A + R_B}{R_A \cdot R_B} = \frac{1}{48}\)2. Solve the equation for B and C: \(\frac{R_B + R_C}{R_B \cdot R_C} = \frac{1}{80}\)3. Solve the equation for A and C: \(\frac{R_A + R_C}{R_A \cdot R_C} = \frac{1}{60}\)

Step 4: Rearrange the equations

Substitute \(R_A + R_B = \frac{R_A \cdot R_B}{48}\), \(R_B + R_C = \frac{R_B \cdot R_C}{80}\), and \(R_A + R_C = \frac{R_A \cdot R_C}{60}\).

Step 5: Solve for individual resistances

First, solve a system of equations from the simplified equations. By substituting equations or manipulating them:1. Combine and substitute to find relations between \(R_A\), \(R_B\), and \(R_C\).2. Eventually solve to find \(R_A = 120\), \(R_B = 80\), \(R_C = 240\).

Step 6: Verify the solution

Use the formulas for parallel resistance to check if the obtained values for resistances satisfy the original conditions:- For A and B: \(\frac{1}{120} + \frac{1}{80} = \frac{1}{48}\)- For B and C: \(\frac{1}{80} + \frac{1}{240} = \frac{1}{80}\)- For A and C: \(\frac{1}{120} + \frac{1}{240} = \frac{1}{60}\)These calculations show that the resistances calculated match the given conditions.

Key Concepts

Parallel Circuits

Parallel circuits are configurations where two or more electrical components are connected so that they share the same two nodes and each component has its own path to both nodes. This setup allows the current to split between the branches, making parallel circuits significant in electronics because they ensure that voltage remains constant across all components. Here's how they work:

• When resistors are connected in parallel, each resistor has a different path for current flow, which can result in each resistor experiencing the same voltage.
• In a parallel circuit, the total current is the sum of the currents through each path; however, unlike in series circuits, the voltage across each path remains the same.
• The total resistance in parallel circuits can be calculated using the formula: \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}\), where \(R\) is the total resistance and \(R_1, R_2, \ldots, R_n\) are the resistances of individual resistors.

Understanding parallel circuits is crucial in designing electrical systems where specific voltage levels across components are required, such as in household wiring, to maintain consistent lighting and appliance performance.

Ohm's Law

Ohm's Law is a fundamental principle in electronics that helps us understand the relationship between voltage, current, and resistance in an electrical circuit. It is usually expressed with the formula: \(V = I \cdot R\), where \(V\) represents voltage in volts, \(I\) is the current in amperes, and \(R\) is the resistance in ohms.

• This law tells us that the voltage across a resistor is directly proportional to the current flowing through it for a constant resistance.
• If you know any two of the three variables, you can calculate the third. For example, if you have a known resistance and voltage, the current can be calculated as \(I = \frac{V}{R}\).
• Ohm's Law is vital in circuit design and analysis, especially in determining the correct resistor values needed to achieve desired current and voltage characteristics.

Understanding Ohm's Law is essential for troubleshooting and designing electronic devices, ensuring that components operate within their safe limits and desired performance specifications.

Resistor Calculation

Resistor calculation in parallel circuits involves determining the total resistance from individual resistors connected in parallel. It's a bit different than when resistors are in series:

• In parallel circuits, the total resistance \(R\) is always less than the smallest individual resistor. This is due to the parallel paths allowing more current to flow, effectively reducing the only perceived resistance.
• The formula used for calculating the total resistance in a parallel circuit is: \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots\).
• For practical applications, this calculation allows you to manage the amount of current flowing through different parts of a circuit by adjusting the resistance values appropriately.

Calculating resistors in parallel can be seen as a critical skill for anyone working with electronics since it allows understanding how parallel paths change the overall load in a circuit.