5 · Parallel circuits

A junction splits the charge

In a parallel circuit, the wire branches at a junction into two or more separate paths, then joins back together further along. Charge arriving at the junction splits between the branches — some goes one way, some the other — and every bit of it eventually gets home, because charge can't just vanish.

Rule 1 Both branches connect the same two points, so both branches feel exactly the same p.d. — whatever the voltage is across one parallel branch, it's the same across the other.

Rule 2 Current splits at the junction and adds back up when the branches rejoin: I_total = I₁ + I₂. The smaller-resistance branch — the easier road — always carries more of it.

Combined resistance goes down

This one surprises people: adding a second parallel path makes it easier overall for current to flow, not harder — because you've literally given the charge a second road to use. So combined parallel resistance is always smaller than the smallest individual branch.

1 / R_p = 1 / R₁ + 1 / R₂

Notice this is the opposite of series, where adding a resistor always makes the total bigger.

Worked example

A 4 Ω resistor and a 12 Ω resistor are connected in parallel across a 6 V supply. Find the current in each branch, and the total current.

Both branches see the full 6 V (Rule 1), so use Ohm's Law on each one separately:

I₁ = V / R₁ = 6 ÷ 4 = 1.5 A · I₂ = V / R₂ = 6 ÷ 12 = 0.5 A

I_total = 1.5 + 0.5 = 2.0 A

Notice Drag either resistance slider and watch the voltmeter reading below — it never moves, because the widget below has no series resistor to eat any voltage first. Every volt the battery gives out appears straight across both branches. That's Rule 1 in its purest form.

Quick check

Q Two resistors, 6 Ω and 3 Ω, are connected in parallel across a 12 V supply. What is the current through the 3 Ω resistor?

The 3 Ω branch sees the full 12 V, same as the 6 Ω branch (Rule 1).

I = V / R = 12 ÷ 3 = 4 A