Litmux Studio

LessonSeries and parallel circuits · 15 min

Parallel circuits

Parts are in parallel when each one is connected across the same two points. Every socket in a house is in parallel with every other: each gets the full mains voltage, and switching one appliance off does not affect the others.

R1, R2 and R3 in parallel across 12 V. Each has the full 12 V across it; the battery supplies the sum of the three branch currents.
R1, R2 and R3 in parallel across 12 V. Each has the full 12 V across it; the battery supplies the sum of the three branch currents.

The three rules of a parallel circuit

  1. The same voltage is across every branch.
  2. Each branch current follows Ohm's law on its own: I = V / R.
  3. The currents add up: the supply provides the total.
1 / Rtotal = 1 / R1 + 1 / R2 + 1 / R3 + ...
Rtotal
the single resistance that would take the same total current - always smaller than the smallest branch
Two resistors: Rtotal = R1 × R2 / (R1 + R2) n equal resistors: R / n
R1 × R2 / (R1 + R2)
"product over sum" - only for two at a time
Worked example

1 kΩ, 2 kΩ and 3 kΩ in parallel across 12 V. Find each current, the total and Rtotal.

  1. I1 = 12 / 1 k = 12 mA, I2 = 12 / 2 k = 6 mA, I3 = 12 / 3 k = 4 mA.
  2. Itotal = 12 + 6 + 4 = 22 mA.
  3. Rtotal = 12 V / 22 mA = 0.545 kΩ = 545 Ω.
  4. Check with the formula: 1 / (1/1 + 1/2 + 1/3) = 1 / 1.833 = 0.545 kΩ.

Answer: 12, 6 and 4 mA; 22 mA in all; 545 Ω - less than the smallest, 1 kΩ.

R1 = 1 kΩ combined with R2: in series and in parallel02468100246810R2 (kΩ)R total (kΩ)R2 = R1
series: R1 + R2parallel: R1 R2 / (R1 + R2)

In parallel, adding a branch always lowers the total resistance - there is one more path for the current. However big R2 gets, the parallel pair never exceeds 1 kΩ.

InteractiveTry it: two resistors in parallel
R1 ∥ R2-Ω
Current in R1-mA
Current in R2-mA
Total current-mA

Make R1 = R2: Rtotal is exactly half.

Circuit for: Try it: two resistors in parallel
The circuit at the starting values

Where you meet it

House wiring

Every socket and light is a branch across the 230 V mains.

Car and motorbike lights

In parallel, so one blown bulb does not take out the others.

Solar panels

Panels in parallel add their currents at the same voltage.

Power resistors

Two 100 Ω 1 W resistors in parallel make 50 Ω that can take 2 W.

Practice 1What is 10 kΩ in parallel with 10 kΩ?

Show the answer

Answer: 5000 Ω

Two equal resistors: R / 2 = 5 kΩ.

Practice 2What is 100 Ω in parallel with 220 Ω?

Hint

Product over sum.

Show the answer

Answer: 68.75 Ω

100 × 220 / 320 = 68.75 Ω.

Practice 3Three lamps take 0.5 A, 0.8 A and 1.2 A from a 12 V battery. What current does the battery supply?

Show the answer

Answer: 2.5 A

In parallel the currents add: 0.5 + 0.8 + 1.2 = 2.5 A.

Challenge 4What resistor must go in parallel with 1 kΩ to make 750 Ω?

Hint

1 / R = 1 / 750 − 1 / 1000.

Show the answer

Answer: 3000 Ω

1 / R = 0.001333 − 0.001 = 0.000333, so R = 3000 Ω (3 kΩ).

Challenge 5An extension board rated 13 A at 230 V already has a 1200 W microwave and a 150 W fridge on it. What is the most power (in W) you can still plug in?

Hint

The board's maximum power is 230 V × 13 A.

Show the answer

Answer: 1640 W

Maximum: 230 × 13 = 2990 W. Already used: 1350 W. Left: 1640 W - not enough for a 2000 W kettle.