EG1216 · Physics 2 — Electricity & Magnetism
Theme 3 · Current & DC circuits

Resistors in series & parallel

Reduce a tangle of resistors to one. Series ones add; parallel ones combine "upside-down" — the mirror image of the capacitor rules.

Source: Serway & Jewett, Physics for Scientists and Engineers, 7th ed., §28.2.

Before you start

What you need first

  • Topic 17 — Ohm's law \(V = IR\).
  • Topic 13 — series/parallel for capacitors (the mirror image).

What you'll be able to do

  • Combine resistors in series: \(R_{\text{eq}} = R_1 + R_2 + \cdots\).
  • Combine resistors in parallel: \(\tfrac{1}{R_{\text{eq}}} = \tfrac{1}{R_1} + \tfrac{1}{R_2} + \cdots\).
  • Know what is shared (current vs voltage).
  • Reduce a simple mixed network to one resistance.

In a line

Resistors in series — add them up

Resistors in a line carry the same current (it has nowhere else to go). In a line is like one longer resistor, so the resistances add:

$$ R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots $$
R₁ R₂
Series: one path, the same current through each.
📐 Worked example 1

Three resistors in series

\(R_1 = 2.0\ \Omega\), \(R_2 = 4.0\ \Omega\), \(R_3 = 6.0\ \Omega\) are in series across a 12 V battery. Find the total resistance and the current.

1Add the resistances:
$$ R_{\text{eq}} = 2.0 + 4.0 + 6.0 = 12\ \Omega $$
2Current from Ohm's law, \(I = V/R_{\text{eq}}\):
$$ I = \frac{12}{12} = 1.0\ \text{A} $$

Side by side

Resistors in parallel — the upside-down rule

Resistors side by side share the same two ends, so they feel the same voltage. More paths means easier flow, so the total resistance drops — below the smallest one:

$$ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots $$
R₁ R₂
Parallel: two paths share the same voltage.
📐 Worked example 2

Two resistors in parallel

\(R_1 = 6.0\ \Omega\) and \(R_2 = 3.0\ \Omega\) are in parallel. Find the equivalent resistance.

1Add the reciprocals:
$$ \frac{1}{R_{\text{eq}}} = \frac{1}{6.0} + \frac{1}{3.0} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} $$
2Flip back:
$$ R_{\text{eq}} = 2.0\ \Omega $$
It is smaller than both 3 Ω and 6 Ω. (Don't stop at \(\tfrac12\) — flip it.)

The mirror of the capacitor rules

Notice the swap: resistors add in series and combine reciprocally in parallel; capacitors do the opposite — they add in parallel and combine reciprocally in series. If you remember one set, flip it for the other.

✏️ Try it yourself

Two \(4.0\ \Omega\) resistors are wired in parallel; that combination is in series with a \(2.0\ \Omega\) resistor, all across a 12 V battery.

(a) Find the parallel pair's resistance.
(b) Find the total resistance.
(c) Find the current drawn from the battery.

(a) \(\dfrac{1}{R_p} = \dfrac{1}{4}+\dfrac{1}{4} = \dfrac{1}{2}\Rightarrow R_p = 2.0\ \Omega.\) (b) In series with 2.0 Ω: \(R_{\text{eq}} = 2.0 + 2.0 = 4.0\ \Omega.\) (c) \(I = V/R_{\text{eq}} = 12/4.0 = 3.0\ \text{A}.\)

Common mistakes

MistakeFix
Adding resistors in parallel.Resistors add in series; in parallel you add reciprocals.
Stopping at \(1/R_{\text{eq}}\) in parallel.Flip it back to get \(R_{\text{eq}}\).
Thinking parallel gives more resistance.Parallel is always less than the smallest resistor.
Assuming the same current in parallel.Parallel shares the voltage; series shares the current.

Recap — the whole topic on one screen

ConnectionRuleShared
Series\(R_{\text{eq}} = R_1 + R_2 + \cdots\) (bigger)same current
Parallel\(\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots\) (smaller)same voltage

Next topic

Kirchhoff's rules

Some circuits can't be reduced by series/parallel alone. Next: two simple rules — one for junctions, one for loops — that crack any circuit, however messy.

→ Topic 21 · Kirchhoff's rules