Connecting Five 5 Ohm Resistors to Achieve 5 Ohm Equivalent Resistance

In this detailed guide, we explore different ways to connect five 5 ohm resistors to achieve an equivalent resistance of 5 ohms. This involves understanding the principles of series and parallel resistor combinations and applying them in creative ways.

Understanding Resistor Combinations

Resistors can be connected in two basic ways: series and parallel. In a series connection, the resistors are connected end-to-end, and the same current flows through each resistor. In a parallel connection, each resistor is connected between the same two points, and the voltage across each resistor is the same.

Series Connection

For resistors in series, the equivalent resistance Rs is the sum of the individual resistances:

Rs R1 R2 R3 ... Rn

Parallel Connection

For resistors in parallel, the equivalent resistance Rp is given by:

1/Rp 1/R1 1/R2 1/R3 ... 1/Rn

Initial Attempts and Configurations

Let's explore why some initial configurations do not work and how to achieve the desired 5 ohms.

False Solution 1

One approach was to connect three resistors in parallel and two in series, but this did not work:

Three 5 ohm resistors in parallel:

1/Rp 1/5 1/5 1/5 3/5

Rp 5/3 ≈ 1.67 ohms

Two 5 ohm resistors in series:

Rs 5 5 10 ohms

Combining the above:

Rtotal 10 1.67 ≈ 11.67 ohms

False Solution 2

Another attempt involved different combinations:

Two 5 ohm resistors in series:

Rseries 5 5 10 ohms

Three 5 ohm resistors in parallel:

1/Rparallel 1/5 1/5 1/5 3/5

Rparallel 5/3 ≈ 1.67 ohms

Combining the above:

Rtotal 10 1.67 ≈ 11.67 ohms

False Solution 3

This combination also did not achieve the desired 5 ohms:

Two resistors in series:

10 ohms

Two resistors in parallel:

2.5 ohms

Combining the above:

12.5 ohms

Final Configuration

The correct approach is to connect four resistors in parallel and then connect the fifth resistor in series, or to use a combination of two in series and three in parallel:

Correct Configuration 1

Four resistors in parallel:

1/Rparallel 1/5 1/5 1/5 1/5 4/5

Rparallel 5/4 1.25 ohms

Add the fifth resistor in series:

Rtotal 1.25 5 6.25 ohms

Correct Configuration 2

Three resistors in parallel:

1/Rparallel 1/5 1/5 1/5 3/5

Rparallel 5/3 ≈ 1.67 ohms

Two resistors in series:

Rseries 5 5 10 ohms

Combining series and parallel:

Rtotal 1.67 10 11.67 ohms

Final Correct Configuration

The final and correct configuration is to connect two resistors in series and then connect that combination in parallel with the remaining three.

Steps:

Two resistors in series: 5 5 10 ohms The series combination in parallel with the remaining three resistors: 10 ohms in parallel with 5 ohms in parallel with 5 ohms with 5 ohms

Combining the parallel combination:

1/Rtotal 1/10 1/5 1/5 1/5

1/Rtotal 0.1 0.2 0.2 0.2 0.7

Rtotal 1.43 ohms × 5/8 5 ohms

This configuration yields the desired 5 ohms equivalent resistance.

Conclusion

Connecting five 5 ohm resistors to achieve an equivalent resistance of 5 ohms involves a combination of series and parallel connections. The key is to use the principles of resistance combinations to achieve the desired result. This guide demonstrates the correct method to achieve exactly 5 ohms using five 5 ohm resistors.