Class 11> Unit # 10: DC Circuits > Kirchhoff's Laws for Series and Parallel Resistors

Kirchhoff’s Laws for Series and Parallel Combination of Resistors — Unit 10

Welcome to Talha's Physics Academy. In this lecture, we apply Kirchhoff's Voltage and Current Laws to analyze series and parallel combinations of resistors and derive the formulas for their equivalent resistance.

Video Lecture

1. Series Combination of Resistors

In a series combination, resistors are connected end-to-end so that there is only a single path for the flow of electric current. The same current \(I\) flows through each resistor, while the total voltage divides across them.

• Kirchhoff's Voltage Law (KVL) Application: The total voltage \(V\) supplied by the source equals the sum of potential differences across individual resistors: $$V = v_1 + v_2$$
• Derivation using Ohm's Law (\(v = IR\)): $$IR = IR_1 + IR_2$$ $$IR = I(R_1 + R_2)$$ $$R = R_1 + R_2$$
• General Formula for \(n\) Resistors in Series: $$R = R_1 + R_2 + R_3 + \dots + R_n$$

2. Parallel Combination of Resistors

In a parallel combination, the corresponding ends of each resistor are connected together at common junctions. The potential difference across each resistor remains the same, while the total current divides among the branches.

• Kirchhoff's Current Law (KCL) Application: Total current entering the junction equals the sum of branch currents: $$I = i_1 + i_2$$
• Derivation using Ohm's Law (\(i = \frac{v}{R}\)): $$\frac{V}{R} = \frac{V}{R_1} + \frac{V}{R_2}$$ $$\frac{V}{R} = V\left(\frac{1}{R_1} + \frac{1}{R_2}\right)$$ $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$
• General Formula for \(n\) Resistors in Parallel: $$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$$ Note: The equivalent resistance in a parallel combination is always smaller than the smallest individual resistor in the network.