Parallel Circuits — Branch Currents (Worked Example)

Given

• R₁ = 30 Ω

• R₂ = 60 Ω

• V = 12 V

• parallel connection

Find

R_eq, I₁, I₂, I_total.

Principle

In parallel, every branch sees the full voltage; conductances add, so the equivalent resistance is less than the smallest branch.

Formula

Solution

1. Combine: $1/R_{eq} = 1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20$

2. Invert: $R_{eq} = 20\ \Omega$

3. Branch currents: $I_1 = 12/30 = 0.40\ \mathrm{A}$; $I_2 = 12/60 = 0.20\ \mathrm{A}$

4. Total: $I = 0.40 + 0.20 = 0.60\ \mathrm{A}$ (check: $12/20 = 0.60\ \mathrm{A}$ ✓)

Unit check

1/Ω + 1/Ω = 1/Ω, inverted to Ω; V/Ω = A. Consistent.

What this means physically

Each branch draws current independently — the 30 Ω path takes twice the current of the 60 Ω path. This is why household outlets are wired in parallel: each appliance gets full voltage regardless of the others.