Parallel Circuits

DC Circuits · 10 min read

Every socket in your house shares the same 230 V supply. Unplug the kettle and the fridge keeps humming — the toaster doesn't care. That independence is the point of a parallel circuit.

Contrast this with a series string: one break stops everything. In parallel, each appliance has its own private path back to the source.

What “Parallel” Means

In a parallel circuit, components share the same two nodes — the same start point and the same end point. There are multiple paths for current, one per branch.

More branches means more paths and less total opposition to current flow.

Both branches share the same top and bottom nodes — same voltage across each.

The Two Golden Rules

Property	Rule
Voltage	Same voltage across every branch
Current	Branch currents add up to the source current

Total Resistance

Parallel resistors combine by their reciprocals — the result is always less than the smallest branch:

\frac{1}{R_{total}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots + \frac{1}{R_{n}}

Two shortcuts come up constantly:

\text{Two branches:}\quad R_{total} = \frac{R_1 R_2}{R_1 + R_2}

N\text{ equal branches:}\quad R_{total} = \frac{R}{N}

Branch Currents

The same voltage $V$ sits across every branch, so each branch current is:

I_{i} = \frac{V}{R_{i}}

The branch with the smallest resistor carries the most current. Double its resistance and its current halves — the other branches are completely unaffected.

Source Current

All branch currents return to the source, so:

I_{source} = I_{1} + I_{2} + \dots + I_{n}

This is Kirchhoff's Current Law (KCL) in action — charge cannot accumulate at a node, so what flows in must flow out.

Worked Example

Three branches — 4 Ω, 6 Ω, and 12 Ω — connected in parallel across 12 V. Find the total resistance, source current, and each branch current.

Find total resistance: $\tfrac{1}{R} = \tfrac{1}{4} + \tfrac{1}{6} + \tfrac{1}{12} = \tfrac{3+2+1}{12} = \tfrac{1}{2}$ → $R_{total} = 2\text{ Ω}$
Find source current: $I_{source} = 12 / 2 = 6\text{ A}$
Find branch currents: $I_1 = 12/4 = 3\text{ A}$ , $I_2 = 12/6 = 2\text{ A}$ , $I_3 = 12/12 = 1\text{ A}$
Verify (KCL): $3 + 2 + 1 = 6\text{ A}$ ✓

Parallel circuits share voltage and split current. The smallest resistor always carries the most current; adding any branch always lowers total resistance.

Think Before Calculating

Predict the answer first — then verify with the parallel circuit rules.

Scenario 1: Two branches — 4 Ω and 12 Ω — are connected in parallel with a 12 V source. What is the total resistance and source current?

Scenario 2: You add a third 6 Ω branch to the circuit above (still 12 V). Does total resistance increase or decrease?

Scenario 3: In a 3-branch parallel circuit, one branch burns open (breaks). What happens to the remaining branches?

Common Mistakes

Parallel Circuit Playground

Drag the sliders — the bar shows how source current divides between branches. The branch with the least resistance takes the most current.

Voltage12.0 V

1 V24 V

R₁4 Ω

R₂6 Ω

R₃12 Ω

R_{total}

2 Ω

I_{source} = I_1 + I_2 + I_3

6.000 A

Current split across each branch

✅ Parallel = multiple paths between the same two nodes
✅ Every branch shares the full supply voltage
✅ Smaller resistor carries more current
✅ Total resistance is always less than the smallest branch
✅ Branch currents add up to source current (KCL)