Calculate the combined resistance of two resistors in parallel

4.3.2 Series and Parallel Circuits

Learning Objective

Calculate the combined resistance of two resistors when they are connected in parallel.

Key Concepts

• Resistance (R) – opposition to the flow of electric current, measured in ohms (Ω).
• Parallel connection – the ends of each resistor are joined to the same two points, so the voltage across each resistor is the same.
• Combined (equivalent) resistance – the single resistance that could replace the parallel network without changing the overall current from the source.

Formula for Two Resistors in Parallel

The reciprocal of the equivalent resistance \$R_{\text{eq}}\$ is the sum of the reciprocals of the individual resistances:

\$\frac{1}{R{\text{eq}}}= \frac{1}{R1}+ \frac{1}{R_2}\$

Solving for \$R_{\text{eq}}\$ gives:

\$R{\text{eq}} = \frac{R1 R2}{R1 + R_2}\$

Derivation (Step‑by‑Step)

1. Because the resistors are in parallel, the voltage \$V\$ across each resistor is the same.
2. Apply Ohm’s law to each resistor: \$I1 = \dfrac{V}{R1}\$ and \$I2 = \dfrac{V}{R2}\$.
3. The total current supplied by the source is the sum of the branch currents: \$I{\text{total}} = I1 + I_2\$.
4. Substitute the expressions for \$I1\$ and \$I2\$:

   \$I{\text{total}} = \frac{V}{R1} + \frac{V}{R2}= V\!\left(\frac{1}{R1}+\frac{1}{R_2}\right)\$

5. Define the equivalent resistance \$R{\text{eq}}\$ by \$I{\text{total}} = \dfrac{V}{R_{\text{eq}}}\$.
6. Equate the two expressions for \$I{\text{total}}\$ and solve for \$R{\text{eq}}\$:

   \$\frac{V}{R{\text{eq}}}= V\!\left(\frac{1}{R1}+\frac{1}{R_2}\right)\$

   \$\frac{1}{R{\text{eq}}}= \frac{1}{R1}+\frac{1}{R_2}\$

   \$R{\text{eq}} = \frac{R1 R2}{R1 + R_2}\$

Worked Example

Two resistors, \$R1 = 120\;\Omega\$ and \$R2 = 80\;\Omega\$, are connected in parallel. Find the equivalent resistance.

1. Write the formula: \$R{\text{eq}} = \dfrac{R1 R2}{R1 + R_2}\$.
2. Substitute the values:

   \$\$R_{\text{eq}} = \frac{120 \times 80}{120 + 80}

   = \frac{9600}{200}

   = 48\;\Omega\$\$

3. Interpretation: The parallel combination offers less resistance than either individual resistor, as expected.

Practice Questions

1. Two resistors, \$R1 = 150\;\Omega\$ and \$R2 = 300\;\Omega\$, are placed in parallel. Calculate \$R_{\text{eq}}\$.
2. A circuit contains a \$220\;\Omega\$ resistor in parallel with an unknown resistor \$Rx\$. The measured equivalent resistance is \$88\;\Omega\$. Find \$Rx\$.
3. Three resistors \$R1 = 100\;\Omega\$, \$R2 = 200\;\Omega\$, and \$R3 = 300\;\Omega\$ are connected in parallel. Determine the total resistance using the two‑resistor formula twice (first combine \$R1\$ and \$R2\$, then combine the result with \$R3\$).

Common Mistakes to Avoid

• Adding resistances directly (that works only for series circuits).
• Forgetting that the voltage across each resistor in parallel is the same.
• Using the series formula \$R{\text{eq}} = R1 + R_2\$ for a parallel arrangement.

Summary Table