Resistors in Parallel

When several resistors are connected between the same two points, they're said to be joined in parallel. Unlike series, every resistor here sees the same potential difference, the one across the shared pair of points, which you can check by placing a voltmeter across each resistor individually.

The current, though, splits between the branches. If I₁, I₂, and I₃ are the currents through three parallel resistors, the total current drawn from the battery is their sum: I = I₁ + I₂ + I₃.

Applying Ohm's law to the combination (I = V/Rp) and to each resistor (I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃), and substituting into I = I₁ + I₂ + I₃, gives V/Rp = V/R₁ + V/R₂ + V/R₃. Since V cancels, 1/Rp = 1/R₁ + 1/R₂ + 1/R₃: the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances. This always makes Rp *smaller* than the smallest individual resistance.

Parallel circuits solve both problems series circuits have: each appliance gets the full voltage and draws exactly the current it needs (a bulb and a heater can coexist happily), and if one component fails, the others keep working, which is exactly how household wiring is done.