Series resistors: deriving R = R₁ + R₂

The main symbols used in this lesson:

Two components are in series when they are connected one after the other along a single path, so the same current has to pass through both of them.

That single path has two immediate consequences, and both come from Kirchhoff's laws.

• Kirchhoff's first law rules out any way for the current to change: there is no junction to divide it, so the current is the same through every component in series.
• Kirchhoff's second law says the p.d.s add up round the loop, so the total p.d. across the pair is the sum of the individual p.d.s.

Connect two resistors $R_1$ and $R_2$ in series with a source of e.m.f. $V$. They behave like one single resistor of resistance $R$, and here is why, step by step.

1. Same current (first law). The path is unbroken, so a current $I$ passes through both.
2. p.d.s from the definition of resistance. $R = V/I$ rearranges to $V = IR$, so $V_1 = IR_1$ and $V_2 = IR_2$.
3. p.d.s add (second law). Going round the loop, $V = V_1 + V_2$.
4. Substitute and match forms. $V = IR_1 + IR_2 = I(R_1 + R_2)$. The pair behaves like one resistor with $V = IR$, so $R = R_1 + R_2$.

The same argument repeats for any number of resistors: $R = R_1 + R_2 + R_3 + \dots$