Derive an expression for the equivalent resistance of resistors connected in parallel. If three resistances of 5 ohms are connected in parallel, calculate the equivalent resistance of the combination.

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Points to Remember:

  • Resistors in parallel offer multiple paths for current flow.
  • The equivalent resistance is always less than the smallest individual resistance.
  • The reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances.

Introduction:

Electrical circuits frequently utilize resistors to control the flow of current. Resistors can be connected in series (one after another) or in parallel (side-by-side). Understanding how to calculate the equivalent resistance of these configurations is crucial in circuit analysis and design. This question requires a factual and analytical approach to derive the formula for equivalent resistance in a parallel configuration and apply it to a specific example.

Body:

1. Deriving the Expression for Equivalent Resistance in Parallel:

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Consider ‘n’ resistors (R₁, R₂, R₃,… Rₙ) connected in parallel across a voltage source ‘V’. According to Ohm’s law, the current (I) flowing through each resistor is given by:

  • I₁ = V/R₁
  • I₂ = V/R₂
  • I₃ = V/R₃
  • …
  • Iₙ = V/Rₙ

The total current (IT) flowing into the parallel combination is the sum of the individual currents:

IT = I₁ + I₂ + I₃ + … + Iₙ

Substituting the expressions for individual currents:

IT = V/R₁ + V/R₂ + V/R₃ + … + V/Rₙ

IT = V (1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ)

Now, let Req be the equivalent resistance of the parallel combination. The total current can also be expressed as:

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IT = V/Req

Equating the two expressions for IT:

V/Req = V (1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ)

Dividing both sides by V:

1/Req = 1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ

Therefore, the equivalent resistance of resistors connected in parallel is given by the reciprocal of the sum of the reciprocals of the individual resistances.

2. Calculating Equivalent Resistance for Three 5-ohm Resistors:

We have three resistors, each with a resistance of 5 ohms (R₁ = R₂ = R₃ = 5 ohms). Using the derived formula:

1/Req = 1/R₁ + 1/R₂ + 1/R₃

1/Req = 1/5 + 1/5 + 1/5 = 3/5

Req = 5/3 ohms

Req ≈ 1.67 ohms

Conclusion:

The equivalent resistance of resistors connected in parallel is always less than the smallest individual resistance. This is because the parallel configuration provides multiple paths for current to flow, effectively reducing the overall resistance. We derived the formula 1/Req = 1/R₁ + 1/R₂ + 1/R₃ + … + 1/Rₙ for calculating the equivalent resistance of ‘n’ resistors in parallel. Applying this formula to the case of three 5-ohm resistors, we found the equivalent resistance to be approximately 1.67 ohms. This understanding is fundamental to circuit design and analysis, ensuring efficient and safe operation of electrical systems. Further research into more complex circuit configurations and the application of Kirchhoff’s laws can enhance understanding of electrical networks.

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