In a series circuit the components sit one after another on a single path, so the same current passes through each of them. In a parallel circuit they sit side by side on separate branches, so the same voltage appears across each of them. Almost every numerical in the chapter follows from those two sentences.
The comparison table
| Series | Parallel | |
|---|---|---|
| Paths for the current | One | One per branch |
| Current | Same through every component | Divides between the branches |
| Voltage | Divides between the components | Same across every branch |
| Equivalent resistance | R = R₁ + R₂ + … | 1/R = 1/R₁ + 1/R₂ + … |
| Size of the equivalent | Larger than the largest resistor | Smaller than the smallest resistor |
| If one component fails | Everything stops | The other branches keep working |
| Typical use | Decorative light strings, a fuse with the device it protects | Household wiring |
One pair of resistors, connected both ways
Take a 6 Ω resistor and a 3 Ω resistor, and a 9 V battery. Every step below is Ohm's law, V = IR. If that relation is not yet automatic, work through Ohm's law explained with examples before going on.
In series
- Equivalent resistance: R = 6 + 3 = 9 Ω.
- Current from the battery: I = V ÷ R = 9 ÷ 9 = 1 A, and that same 1 A flows through both resistors.
- Voltage across the 6 Ω resistor: 1 × 6 = 6 V. Across the 3 Ω resistor: 1 × 3 = 3 V.
Check: 6 V + 3 V = 9 V, the battery voltage. In series the larger resistor takes the larger share of the voltage.
In parallel
- Equivalent resistance: 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6, so R = 2 Ω.
- Each branch has the full 9 V across it.
- Current in the 6 Ω branch: 9 ÷ 6 = 1.5 A. In the 3 Ω branch: 9 ÷ 3 = 3 A.
- Total current from the battery: 1.5 + 3 = 4.5 A.
Check: 9 V ÷ 2 Ω = 4.5 A. In parallel the smaller resistor carries the larger share of the current.
The same two resistors draw 1 A in one arrangement and 4.5 A in the other. The arrangement, not just the parts, decides what the battery has to supply.
A shortcut for two resistors in parallel
For exactly two resistors, the reciprocal formula rearranges to product over sum:
R = (R₁ × R₂) ÷ (R₁ + R₂)
For 6 Ω and 3 Ω that gives 18 ÷ 9 = 2 Ω, the same answer with less fraction work. For n equal resistors of value R in parallel the result is simply R ÷ n. Do not stretch the product-over-sum form to three resistors; it only holds for two.
Why homes are wired in parallel
If the lights, the fan and the refrigerator were in series, they would have to share the supply voltage, none would run at its rated value, and switching off one would switch off all of them. Wired in parallel, each gets the full supply voltage and each has its own switch.
The cost is that every appliance you turn on adds its current to the total drawn from the mains. Too many at once and the wiring overheats, which is the situation fuses and circuit breakers exist to stop. The fuse itself is placed in series with the circuit it protects, so that when it melts, the current stops.
Where marks are lost
- Leaving a parallel answer as 1/R. After adding the reciprocals you must invert. 1/R = 1/2 means R = 2 Ω, not 0.5 Ω.
- A parallel result bigger than a branch. If your equivalent is larger than the smallest resistor, something went wrong.
- Giving every series resistor the full battery voltage. In series the voltage is shared. In parallel it is not.
- Mixed circuits done in one step. Reduce the innermost series or parallel group first, redraw, then repeat.
Quick check
Three 12 Ω resistors are connected first in series and then in parallel. Find the equivalent resistance each time.
Series: 12 + 12 + 12 = 36 Ω. Parallel: 12 ÷ 3 = 4 Ω.