Calculate the combined resistance of two resistors in parallel

⚡️ 4.3.2 Series and Parallel Circuits

Objective: Calculate the combined resistance of two resistors in parallel

In a parallel circuit, the current has multiple paths to travel, just like water flowing through several pipes that join back together. The total resistance is always lower than any single resistor, which means the circuit is easier for the current to pass through.

Analogy 🚗: Imagine two roads (resistors) leading to the same destination. If you can choose either road, the overall “traffic resistance” is reduced because traffic can split. The more roads you add, the easier it is to get through.

Formula

The combined resistance \(R{\text{eq}}\) of two resistors \(R1\) and \(R_2\) in parallel is:

\( \displaystyle \frac{1}{R{\text{eq}}} = \frac{1}{R1} + \frac{1}{R_2} \)

or, solved for \(R_{\text{eq}}\):

\( \displaystyle R{\text{eq}} = \frac{R1 R2}{R1 + R_2} \)

Step‑by‑Step Example

  1. Take \(R1 = 10\,\Omega\) and \(R2 = 15\,\Omega\).

  2. Multiply the resistances: \(10 \times 15 = 150\).

  3. Add the resistances: \(10 + 15 = 25\).

  4. Divide the product by the sum: \(150 \div 25 = 6\).

  5. Result: \(R_{\text{eq}} = 6\,\Omega\).

Quick Formula Box

\( R{\text{eq}} = \dfrac{R1 R2}{R1 + R_2} \)

Exam Tips 🧠

  • Always check units – resistances are in Ω.

  • Use the reciprocal form first if you’re comfortable with fractions.

  • Remember that the combined resistance of parallel resistors is always lower than the smallest individual resistor.

  • For quick calculations, you can use the shortcut: \(R{\text{eq}} = \frac{R1 R2}{R1 + R_2}\).

  • Practice with different values to build confidence.

Practice Problem 🔍

Calculate the equivalent resistance of \(R1 = 8\,\Omega\) and \(R2 = 12\,\Omega\) in parallel.

StepCalculation
1. Multiply\(8 \times 12 = 96\)
2. Add\(8 + 12 = 20\)
3. Divide\(96 \div 20 = 4.8\)
Result\(R_{\text{eq}} = 4.8\,\Omega\)

Key Takeaway ??

Parallel circuits let current split, which reduces the overall resistance. Use the reciprocal formula or the product‑over‑sum shortcut to find \(R_{\text{eq}}\).