Problem 54 - Algebra 2

Question

The total resistance \(R\) for a parallel circuit with three bulbs is \(R=\frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}}\). a. Simplify the right side of the equation. b. Find the total resistance \(R\) of a parallel circuit with bulbs that have resistances of 3 ohms, 2.5 ohms, and 2.5 ohms.

Step-by-Step Solution

Answer

a) The simplified resistance formula for a parallel circuit with three resistors is \(R_{total} = R_{1}R_{2}R_{3}/(R_{2}R_{3} + R_{1}R_{3} + R_{1}R_{2})\).\nb) The total resistance of the parallel circuit with bulbs of 3 ohms, 2.5 ohms, and 2.5 ohms is 0.88 ohms.

Step 1: Simplify the Equation.

Begin by rewriting the equation without the denominator into the form \(R_{total} = 1/(1/R_{1} + 1/R_{2} + 1/R_{3})\) which simplifies to \(R_{total} = R_{1}R_{2}R_{3}/(R_{2}R_{3} + R_{1}R_{3} + R_{1}R_{2})\). This is the simplified form of the mathematical equation.

Step 2: Substitute the values.

Given that \(R_1=3 \, ohms\), \(R_2=2.5 \, ohms\), and \(R_3=2.5 \, ohms\) are resistances of three bulbs. Substitute these values into the formula so \(R=3*2.5*2.5/(2.5*2.5 + 3*2.5 + 3*2.5)\).

Step 3: Perform the calculations.

When you perform the calculations, you get that \(R = 18.75 / (6.25 + 7.5 + 7.5) = 18.75 / 21.25\).

Step 4: Final Calculation.

Upon final calculation, you get \(R = 0.88 \, ohms\). So, the total resistance of the parallel circuit with these bulbs is 0.88 ohms.

Key Concepts

Understanding Resistance Calculation in Parallel Circuits

In a parallel circuit, the concepts of resistance calculation are slightly different compared to series circuits. When components like bulbs are connected in parallel, each bulb provides a separate path for the current. This configuration reduces the total resistance of the circuit.
The formula for calculating the total resistance \(R\) in a parallel circuit with three resistors is:

• \(R = \frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}} \)

This might look complex at first, but it essentially means you're calculating the reciprocal of the sum of reciprocals of individual resistances.
Breaking down the process, we take the reciprocal of each resistance, add them together, and finally take the reciprocal of that sum. Essentially, the parallel arrangement brings down the effective resistance making it always less than the smallest individual resistance among the components.
Understanding this concept is key to troubleshooting and optimizing electrical circuits involving parallel connections.

Mastering Simplification of Equations

Simplifying the resistance equation for parallel circuits helps in faster calculations and better understanding. Starting from \(R = \frac{1}{\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}} \), we aim to get the equation into a form that's easier to handle.
First, you identify that you can simplify using the Least Common Denominator (LCD) approach. The equation becomes:

• \( R_{total} = \frac{R_{1}R_{2}R_{3}}{R_{2}R_{3} + R_{1}R_{3} + R_{1}R_{2}} \)

This expression is derived by multiplying all terms by the product of the resistances \( R_1R_2R_3 \) to eliminate the fractions inside the reciprocal function. This method not only simplifies computations but also reveals a common structure found in parallel resistances.
Being skilled in simplifying these equations can significantly speed up your analysis of electrical circuits and make complex problems easier to digest.

Analyzing Parallel Circuit Properties

Circuit analysis becomes crucial when dealing with multiple electrical paths in parallel circuits. These circuits have unique features compared to series circuits:

• Each component operates independently, allowing simultaneous paths for current flow.
• Total current in the circuit is the sum of currents through individual paths.
• Voltage across each component in a parallel circuit remains the same.

These characteristics make parallel circuits especially useful in our homes where multiple appliances need to operate at the same voltage level.
Understanding how to analyze these circuits involves calculating total resistance, as explored earlier, and recognizing how changes in one path affect the total current and resistance. By mastering circuit analysis, you can design, evaluate, and troubleshoot complex network systems with confidence and accuracy.