Series & Parallel Circuits

You’ll be able to

Combine resistors in series and in parallel into a single equivalent resistance
Determine how current and voltage distribute in series and parallel branches
Reason about what happens to a circuit when a component is added or removed

Series: one path, shared current

Resistors in series sit end to end on a single path, so the same current flows through each one. Their resistances simply add: R = R₁ + R₂ + …, so the total is always larger than any single resistor. The battery’s voltage divides among them in proportion to their resistances (the biggest resistor drops the most voltage). Break a series circuit anywhere — one burned-out bulb — and the whole current stops, which is why old holiday light strings all went dark at once.

Parallel: shared voltage, split current

Resistors in parallel connect across the same two points, so each feels the same voltage — the battery’s full voltage. The current splits among the branches, with more going through the smaller resistance. The equivalent resistance is found by adding reciprocals: 1/R = 1/R₁ + 1/R₂ + …, and it is always smaller than the smallest branch, because adding another path gives charge more ways through. Household outlets are wired in parallel so each device gets the full 120 V and can switch off independently.

Combining resistors

Series: R = R₁ + R₂ + … · Parallel: 1/R = 1/R₁ + 1/R₂ + …

Series total is bigger than any part; parallel total is smaller than any part. For two parallel resistors, a handy shortcut is R = (R₁R₂)/(R₁ + R₂).

Worked example

A 3.0 Ω and a 6.0 Ω resistor are connected in parallel. Find the equivalent resistance.

1.Add reciprocals: 1/R = 1/3.0 + 1/6.0.
2.Common denominator: 1/R = 2/6 + 1/6 = 3/6 = 1/2.
3.Invert: R = 2.0 Ω.
4.Check with the product-over-sum shortcut: (3 × 6)/(3 + 6) = 18/9 = 2.0 Ω.

Answer: R = 2.0 Ω — smaller than either resistor, as every parallel combination must be.

Checkpoint

A 3.0 Ω resistor and a 6.0 Ω resistor are connected in parallel. What is their equivalent resistance?

Watch out

Do not forget the final reciprocal step in the parallel formula. Computing 1/R₁ + 1/R₂ gives you 1/R, not R. Students routinely stop early and report 1/2 Ω instead of flipping it to 2 Ω.

Checkpoint

A 4.0 Ω resistor and an 8.0 Ω resistor are connected in series across a 12 V battery. What is the current from the battery?

On the exam

Anchor each rule with its shared quantity: series shares current, parallel shares voltage. From that, use V = IR branch by branch. Complex networks reduce to one equivalent resistor by collapsing series and parallel groups one step at a time.

Answer the 2 checkpoints as you read.