derive, using Kirchhoff’s laws, a formula for the combined resistance of two or more resistors in parallel

Kirchhoff’s Laws

Objective

Derive, using Kirchhoff’s laws, a formula for the combined resistance of two or more resistors in parallel.

Key Concepts

• Kirchhoff’s Current Law (KCL): The algebraic sum of currents entering a junction is zero. \$\sum I{\text{in}} = \sum I{\text{out}}\$
• Kirchhoff’s \cdot oltage Law (K \cdot L): The algebraic sum of potential differences around any closed loop is zero. \$\sum V = 0\$
• Ohm’s Law: For a resistor, \$V = IR\$.

Derivation for Two Resistors in Parallel

1. Consider a simple parallel circuit with a voltage source \$V\$ and two resistors \$R1\$ and \$R2\$ connected across the same two nodes.
2. Apply KCL at the top node (junction A):

   \$I{\text{source}} = I1 + I_2\$

   where \$I1\$ and \$I2\$ are the currents through \$R1\$ and \$R2\$ respectively.

3. Using Ohm’s law for each branch:

   \$I1 = \frac{V}{R1}, \qquad I2 = \frac{V}{R2}\$

4. Substitute these expressions into the KCL equation:

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

5. Define the equivalent (combined) resistance \$R{\text{eq}}\$ of the parallel network such that \$I{\text{source}} = \dfrac{V}{R_{\text{eq}}}\$.
6. Equate the two expressions for \$I_{\text{source}}\$:

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

7. Cancel \$V\$ (non‑zero) and solve for \$R_{\text{eq}}\$:

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

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

Extension to \$n\$ Resistors in Parallel

For \$n\$ resistors \$R1, R2, \dots , R_n\$ connected in parallel, the same reasoning gives:

\$\frac{1}{R{\text{eq}}} = \sum{k=1}^{n}\frac{1}{R_k}\$

or equivalently,

\$R{\text{eq}} = \left(\sum{k=1}^{n}\frac{1}{R_k}\right)^{-1}\$

Summary Table