Resolving Resistance Values in Parallel and Series Configurations

Understanding the behavior of resistors in different circuit configurations is fundamental in electrical engineering and electronics. One common scenario involves determining the individual resistances of two resistors given their combined resistances when connected in parallel and series. In this article, we'll explore a specific example where two resistors, denoted as R1 and R2, have a net resistance of 3 ohms when connected in parallel, and a combined resistance of 16 ohms when connected in series. We'll walk through the mathematical process to find the values of R1 and R2, and discuss the practical implications of these calculations.

Introduction to Parallel and Series Resistance

When resistors are connected in a parallel circuit, the total resistance R_total is calculated using the formula:

[ frac{1}{R_{total}} frac{1}{R_1} frac{1}{R_2} ]

When resistors are connected in a series circuit, the total resistance is simply the sum of the individual resistances:

[ R_{total} R_1 R_2 ]

In our scenario, we are given that the parallel resistance is 3 ohms and the series resistance is 16 ohms. This problem can be solved mathematically to find the individual values of R1 and R2.

Formulating Equations

Based on the given conditions, we can write two equations:

1. For parallel connection:

[ frac{1}{R_1} frac{1}{R_2} frac{1}{3} ]

2. For series connection:

[ R_1 R_2 16 ]

These are the two equations we need to solve to find the values of R1 and R2.

Solving the Equations

To solve the equations, we start by manipulating the first equation:

[ frac{1}{R_1} frac{1}{R_2} frac{1}{3} ]

First, we find a common denominator for the fractions:

[ frac{R_2 R_1}{R_1R_2} frac{1}{3} ]

By cross-multiplying, we get:

[ 3(R_2 R_1) R_1R_2 ]

Substituting R2 16 - R1 from the second equation into the first equation, we get:

[ 3R_2 3R_1 R_1R_2 ]

We already know that R2 16 - R1, so substituting this in:

[ 3(16 - R_1) 3R_1 R_1(16 - R_1) ]

Expanding and simplifying:

[ 48 - 3R_1 3R_1 16R_1 - R_1^2 ]

[ 48 16R_1 - R_1^2 ]

Arranging the terms to form a standard quadratic equation:

[ R_1^2 - 16R_1 48 0 ]

Factoring the quadratic equation:

[ (R_1 - 4)(R_1 - 12) 0 ]

This gives us two possible solutions:

[ R_1 4 quad text{or} quad R_1 12 ]

Correspondingly, R2 values will be:

If R1 4 ohms, then R2 12 ohms, and vice versa.

Practical Implications

The values of R1 and R2 have practical significance in electrical and electronic circuits. For example, in a parallel circuit with these resistances, each resistor will allow more current to flow than in a series circuit due to the lower effective resistance.

In series, the total resistance is 16 ohms, meaning the voltage drop across the circuit will be higher. However, in parallel, the voltage drop across each resistor will be the same as the supply voltage, which can be beneficial in applications requiring a consistent voltage across different components.

Conclusion

This problem illustrates the importance of understanding how resistance values behave in different circuit configurations. By using the equations for series and parallel resistances, we can accurately determine the individual resistance values, which is crucial for designing and troubleshooting circuits.

To summarize, the values of resistors R1 and R2, given the conditions stated, are 4 ohms and 12 ohms. This highlights the practical implications of electrical configuration and the necessity of accurate calculations in circuit design.