Resistors in series and parallel

In series (one after another, in a single line):

• the same current flows through each resistor;
• the potential differences add up: V = V₁ + V₂ + V₃;
• so the total resistance is the sum: Rₛ = R₁ + R₂ + R₃, bigger than any single one.

In parallel (side by side, each across the same two points):

• each resistor has the same potential difference;
• the currents add up: I = I₁ + I₂ + I₃;
• so 1/Rₚ = 1/R₁ + 1/R₂ + 1/R₃, and the total is smaller than the smallest one. More paths make it easier for current to flow.

Example from the book: 5 Ω, 10 Ω and 30 Ω in parallel on 12 V. Each gets 12 V, so the currents are 12/5 = 2.4 A, 12/10 = 1.2 A and 12/30 = 0.4 A, a total of 4 A. Then Rₚ = 12/4 = 3 Ω: smaller than 5 Ω, as it should be.

Mixed circuits: first replace each parallel group by its single equivalent resistance, then add the series parts. For example, a group worth 8 Ω in series with 10 Ω gives 18 Ω.

"Three 6 Ω resistors: how do you make 4 Ω?" Try the possibilities: all in series gives 18 Ω; all in parallel gives 2 Ω; two in parallel plus one in series gives 9 Ω; two in series with the third across them gives 1/R = 1/12 + 1/6, so 4 Ω.

Why are homes wired in parallel? With series wiring, if one bulb fuses, everything goes off, like a string of old fairy lights, and all the devices would share one current. In parallel, each device gets the full voltage, draws the current it needs, and can be switched on and off on its own.