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

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Kirchhoff’s Laws

Kirchhoff’s Laws

Objective

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

1. Statement of the Laws

  • Kirchhoff’s Current Law (KCL): At any junction, the algebraic sum of currents entering the junction equals the sum of currents leaving it.

    \$\sum I{\text{in}} = \sum I{\text{out}}\$

  • Kirchhoff’s \cdot oltage Law (K \cdot L): Around any closed loop, the algebraic sum of the potential differences (voltage drops and rises) is zero.

    \$\sum V = 0\$

2. Series Connection of Resistors

Consider a simple series circuit powered by a single ideal battery of emf \$V\$ and internal resistance negligible. The circuit contains \$n\$ resistors \$R1, R2, \dots , R_n\$ as shown in the suggested diagram.

Suggested diagram: A battery of emf \$V\$ connected to a series chain of resistors \$R1, R2, \dots , R_n\$ with a single current \$I\$ flowing clockwise.

3. Derivation Using Kirchhoff’s \cdot oltage Law

  1. Identify a closed loop that includes the battery and all resistors.

  2. Apply K \cdot L around the loop. Taking the rise across the battery as \$+V\$ and each voltage drop across a resistor as \$-V_i\$, we have

    \$V - V1 - V2 - \dots - V_n = 0.\$

  3. Use Ohm’s law for each resistor: \$Vi = I Ri\$ (the same current \$I\$ flows through every resistor in series).

  4. Substitute the Ohm’s law expressions into the K \cdot L equation:

    \$V - I R1 - I R2 - \dots - I R_n = 0.\$

  5. Factor out the common current \$I\$:

    \$V = I\,(R1 + R2 + \dots + R_n).\$

  6. Define the equivalent (combined) resistance \$R{\text{eq}}\$ of the series arrangement such that \$V = I R{\text{eq}}\$. Comparing with the previous line gives

    \$R{\text{eq}} = R1 + R2 + \dots + Rn = \sum{k=1}^{n} Rk.\$

4. Key Result

The combined resistance of resistors connected in series is simply the arithmetic sum of the individual resistances:

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

5. Summary Table

LawMathematical FormPhysical Meaning
Kirchhoff’s Current Law (KCL)\$\displaystyle\sum I{\text{in}} = \sum I{\text{out}}\$Charge is conserved at a junction; no accumulation of charge.
Kirchhoff’s \cdot oltage Law (K \cdot L)\$\displaystyle\sum V = 0\$The net change in electric potential around a closed loop is zero.
Series Resistance\$\displaystyle R{\text{eq}} = \sum{k=1}^{n} R_k\$Resistors share the same current; total voltage drop is the sum of individual drops.

6. Common Misconceptions

  • Assuming the current changes between series resistors – in a series circuit the current is identical through each component.

  • Confusing series with parallel combinations – in parallel the reciprocal of the equivalent resistance is the sum of reciprocals.

  • Neglecting the internal resistance of the source when applying K \cdot L – for idealised derivations it is omitted, but in real circuits it must be included.

7. Extension Question

Using the same method, derive the formula for the equivalent resistance of two resistors \$R1\$ and \$R2\$ connected in parallel.