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.
Formula
The combined resistance \(R_{\text{eq}}\) of two resistors \(R_1\) and \(R_2\) in parallel is:
\( \displaystyle \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} \)
or, solved for \(R_{\text{eq}}\):
\( \displaystyle R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2} \)
Step‑by‑Step Example
- Take \(R_1 = 10\,\Omega\) and \(R_2 = 15\,\Omega\).
- Multiply the resistances: \(10 \times 15 = 150\).
- Add the resistances: \(10 + 15 = 25\).
- Divide the product by the sum: \(150 \div 25 = 6\).
- Result: \(R_{\text{eq}} = 6\,\Omega\).
Quick Formula Box
\( R_{\text{eq}} = \dfrac{R_1 R_2}{R_1 + R_2} \)
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: \(R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}\).
- Practice with different values to build confidence.
Practice Problem 🔍
Calculate the equivalent resistance of \(R_1 = 8\,\Omega\) and \(R_2 = 12\,\Omega\) in parallel.
| Step | Calculation |
|---|---|
| 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 \(R_{\text{eq}}\).
Revision
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