B.5.13—Series and parallel circuits

Series rules

In series, the same current passes through each component:

$$I=I_1=I_2$$

Potential differences and resistances add: $$V=V_1+V_2$$ and $$R_s=R_1+R_2$$.

Parallel rules

In parallel, each branch has the same potential difference:

$$V=V_1=V_2$$

Currents add at the junction and reciprocal resistances add:

$$I=I_1+I_2,\quad \frac1{R_p}=\frac1{R_1}+\frac1{R_2}$$

Solve systematically

Identify junctions and branches, replace simple groups with equivalent resistance, then use Ohm’s law and conservation rules to recover branch currents and voltage drops.

Worked comparison from the mapped local textbook

For $5.0\,\mathrm{k\Omega}$ and $8.0\,\mathrm{k\Omega}$ resistors:

$$R_s=5.0+8.0=13\,\mathrm{k\Omega}$$

$$R_p=\left(\frac1{5000}+\frac1{8000}\right)^{-1}=3.1\,\mathrm{k\Omega}$$

The parallel equivalent is smaller than either branch resistance, which is a useful check.

Common trap

Do not use the series current rule in a parallel branch or add parallel resistances directly.