Four resistors each of resistance 2 ohm are joined end to end to form a square abcd

General
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four resistors each of resistance 2 ohm are joined end to end to form a square abcd

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How to Calculate Equivalent Resistance in a Square Circuit of Four 2-Ohm Resistors

When four resistors with resistance equal to 2 ohms each are joined end to end to form a square, we can explore several possible configurations for calculating the equivalent resistance between different points. Let’s delve into the setup and mathematical solutions involved with such a circuit:

Understanding the Configuration

Circuit Layout:

Consider a square circuit formed by four resistors, labeled as follows:

  • Resistor 1: between points A and B
  • Resistor 2: between points B and C
  • Resistor 3: between points C and D
  • Resistor 4: between points D and A

Each resistor has a resistance of 2 ohms.

Calculating Equivalent Resistance Between Diagonal Corners

Scenario 1: Calculate the equivalent resistance across one diagonal, say AC.

In a square setup, two of the resistors will be in series, and the two diagonals we consider will form two parallel paths.

[
\text{Combination 1 (AC): } R_1 \text{ in series with } (\text{parallel of } R_2 \text{ + } R_3) \text{ and } R_4
]

Let’s break down the steps:

  1. Path 1 (AC):

    • Path: A → B → C (Resistors R1 + R2)
    • Total Resistance in Path 1: (R_1 + R_2 = 2 + 2 = 4 , \text{ohms})

  2. Path 2 (Shortcut AC):

    • Path directly from A to C diagonally (Resistor positioned directly)
    • Total Resistance: 2 ohms

  3. Equivalent Resistance:

    • These two resistors are in parallel, so we use the parallel formula:
      [
      \frac{1}{R_{\text{eq}}} = \frac{1}{4} + \frac{1}{2}
      ]
      Solving this gives:
      [
      \frac{1}{R_{\text{eq}}} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}
      ]
      [
      R_{\text{eq}} = \frac{4}{3} \approx 1.33 , \text{ohms}
      ]

Calculating Equivalent Resistance Adjacent Points

Scenario 2: Calculate the equivalent resistance across adjacent points, say AB.

Both the resistors directly between these points are directly in series, as they are lying one after the other without any branch.

  1. Path AB:
    • Directly: Resistor R1
    • Combined with opposite: Resistor R4 (since resistors R2 and R3 form another series)
    • Total Resistance for just two is: ( R_1 + R_4 = 2 + 2 = 4 , \text{ohms}).

So the equivalent resistance for adjacent points (AB, BC, CD, or DA) is 4 ohms.

Conclusion

By understanding how the resistors are connected, you can readily determine the resistance:

  • Diagonal Points (e.g., AC or BD):

    • Equivalent Resistance: ( \approx 1.33 , \text{ohms} )

  • Adjacent Points (e.g., AB, BC, CD, or DA):

    • Equivalent Resistance: ( 4 , \text{ohms} )

The calculations depend on basic principles of circuits like series and parallel combinations. This makes solving resistor networks intuitive once the paths of current through the circuit are analyzed.

This approach ensures a comprehensive understanding of the resistance properties involved in such a square formed by resistors. If you have any additional questions or need further clarifications, feel free to ask! @username