Parallel Circuit

In a parallel circuit components are connected across the same two points, giving multiple paths for current. Voltage is the same across all branches but currents add up.

What Is a Parallel Circuit?

In a parallel circuit, components are connected side-by-side. Each component has its own path for current, all sharing the same two nodes.

Battery and two resistors in parallel branches — current splits between the branches.

Key Rules

Voltage — Parallel

V_1 = V_2 = V_3 = V_{supply}

All branches have the same voltage.

Current — Parallel

I_{total} = I_1 + I_2 + I_3 + \ldots

Total current equals the sum of branch currents.

Total Resistance — Parallel

\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots

Total resistance is always less than the smallest branch resistance.

Note If one branch in a parallel circuit breaks, the others keep working — this is how house wiring is designed.

Two Resistors in Parallel — shortcut

$R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}$

R₁ = 6 Ω and R₂ = 12 Ω in parallel with a 12 V supply. Find total resistance and each branch current.

$R_{total} = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\,\Omega$

$I_1 = \frac{12}{6} = 2 \text{ A}, \quad I_2 = \frac{12}{12} = 1 \text{ A}, \quad I_{total} = 3 \text{ A}$

Real world: Home electrical sockets

Every socket in your home is on a parallel branch of the mains circuit. Each device — a kettle, laptop, TV — receives the full 230 V supply voltage regardless of what else is plugged in. Switching off one appliance does not affect any other, and the total current drawn from the supply equals the sum of all individual appliance currents.

Real world: Hospital life-support equipment

Critical hospital equipment (ventilators, heart monitors, infusion pumps) is always wired in parallel so that a fault or trip in one device cannot cut power to the others. Each machine gets the full supply voltage independently — a failure in one branch does not endanger other patients.

Tip Where do you see parallel circuits in real life? Every socket in your home, every USB hub, car headlights, and solar panel arrays are all examples of parallel circuit design!

Practice Problems

Two 10 Ω resistors are in parallel. What is the total resistance?

Show answer

R_total = (10×10)/(10+10) = 5 Ω.

A parallel circuit has branches drawing 3 A and 5 A. What is the total current?

Show answer

I_total = 3 + 5 = 8 A.