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

Kirchhoff’s Laws: Series Resistors

What are Kirchhoff’s Laws?

⚡️ Imagine a city’s power grid. Kirchhoff’s laws are the rules that keep the electricity flowing smoothly. They are:

• KCL (Current Law) – The total current entering a junction equals the total current leaving it.
• KVL (Voltage Law) – The sum of voltage rises and drops around any closed loop is zero.

Analogies to Understand the Laws

• Water Flow (KCL): If water enters a junction of pipes, the same amount must leave – no water is lost.

• Water Pressure (KVL): Going around a loop, the total pressure gained (from pumps) minus the pressure lost (through friction) equals zero.

Deriving the Series Resistance Formula

Consider two resistors, \(R1\) and \(R2\), connected end‑to‑end (series). The same current \(I\) flows through both.

By Ohm’s law, the voltage across each resistor is \(V1 = I R1\) and \(V2 = I R2\).

Using KVL around the loop:

\$V{\text{source}} = V1 + V2 = I R1 + I R2 = I (R1 + R_2).\$

The total resistance \(R{\text{total}}\) is defined by \(V{\text{source}} = I R_{\text{total}}\).

Therefore:

\$\boxed{R{\text{total}} = R1 + R_2}.\$

For more than two resistors, the same reasoning gives:

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

Practical Example

Two resistors: \(R1 = 5\,\Omega\) and \(R2 = 10\,\Omega\).

Using the formula:

\$R_{\text{total}} = 5\,\Omega + 10\,\Omega = 15\,\Omega.\$

The current through the circuit is the same in both resistors.

Quick Reference Table

Examination Tips

1. Always check that all resistors in a series loop carry the same current.
2. When applying KVL, write the voltage rises first, then the drops.
3. Remember the simple addition rule: \(R{\text{total}} = \sum Rk\). No fractions or reciprocals needed for series.
4. Use the water‑flow analogy to explain why currents are equal in series.
5. Practice converting a circuit diagram to a loop equation before solving.

Quick Quiz

If three resistors of 2 Ω, 4 Ω, and 6 Ω are connected in series, what is the total resistance?

Answer: \(2 + 4 + 6 = 12\,\Omega\). ??