Solving for R2 and R3 in a Circuit with Given Series and Parallel Values

Understanding the values of R2 and R3 given the known values of R1 can be a challenging yet rewarding task for anyone studying electrical circuits. This article will walk you through the process step by step, including the use of series and parallel resistance calculations.

Problem Statement

Given:

The value of R1 is 100 ohms. The total series resistance is 640 ohms. The total parallel resistance is 52 ohms.

What are the values of R2 and R3?

Step-by-Step Solution

First, let's define the known values:

R1  100 ohmsR1   R2   R3  640 ohms1/100   1/R2   1/R3  1/52

From the first known value:

R2 R3 640 - 100 540 ohms.

Next, we use the parallel resistance formula:

( frac{1}{R_{total}} frac{1}{R1} frac{1}{R2} frac{1}{R3} )

Rearranging for R2 and R3:

1/52 - 1/100  1/R2   1/540 - R2

Let's isolate R2 and solve for R3 by substituting R3 540 - R2:

( frac{1}{52} - frac{1}{100} frac{1}{R2} frac{1}{540 - R2} )

Simplifying the left side:

frac{1}{52} - frac{1}{100}  frac{50 - 52}{5200}  -frac{2}{5200}  -frac{1}{2600}

Substituting back into the equation:

( -frac{1}{2600} frac{1}{R2} frac{1}{540 - R2} )

Multiplying through by (2600)(540 - R2)R2 to clear the denominators:

( -R2(540 - R2) 2600R2 2600(540 - R2) )

Expanding and simplifying:

( -540R2 R2^2 2600R2 1404000 - 2600R2 )

Combining like terms:

( R2^2 - 540R2 1404000 )

Forming a quadratic equation:

( R2^2 - 540R2 - 1404000 0 )

Solving the quadratic equation using the quadratic formula:

( R2 frac{-b pm sqrt{b^2 - 4ac}}{2a} )

Here, a 1, b -540, and c -1404000:

( R2 frac{540 pm sqrt{540^2 4 cdot 1404000}}{2} )

( R2 frac{540 pm sqrt{291600 5616000}}{2} )

( R2 frac{540 pm sqrt{5907600}}{2} )

( R2 frac{540 pm 2430}{2} )

( R2 150 , text{or} , 390 , text{Ohms} )

Therefore, the values of R2 and R3 are 150 Ohms and 390 Ohms, respectively.

Verification

To verify, we can calculate the parallel resistance for the two sets of values:

When R2 150 Ohms and R3 390 Ohms: ( 1/52 frac{1}{100} frac{1}{150} frac{1}{390} ) Calculate 1/52 1/150 1/390 and verify it equals 1/52: ( 1/52 1/150 1/390 frac{390 150 100}{52 cdot 150 cdot 390} frac{640}{52 cdot 150 cdot 390} frac{1}{52} ) Thus, the values are correct.

Both values satisfy the given conditions and solve the problem.

Conclusion

Determining the values of resistors in a circuit is crucial for understanding and designing electrical systems. This problem demonstrates the application of series and parallel resistance calculations and solving quadratic equations.

By understanding these concepts, you can apply the same principles to other electrical circuits and homework problems. Remember to always double-check your work to ensure accuracy.