🔌Parallel Circuits
Resistors in Parallel
In a parallel circuit, components are connected side-by-side, providing multiple paths for current.
Key rules for parallel circuits: 1. Voltage is the same across all components: V₁ = V₂ = V₃ = ... 2. Current divides: I_total = I₁ + I₂ + I₃ + ... 3. Resistance decreases: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...
Adding parallel resistors always DECREASES total resistance — more paths means less total opposition.
Current divider: more current flows through lower resistance branches. Smaller R → more current.
Your home wiring is parallel — each device gets the same 120V, and each can be switched independently.
✏️ Worked Example
Problem: Two resistors (6Ω and 3Ω) are connected in parallel with a 12V source. Find R_total, total current, and current through each.
📐 Key Equations
Parallel circuits
(1)/(R_parallel) = (1)/(R_1) + (1)/(R_2) + ·sV_1 = V_2 = V_3 = ·sI_total = I_1 + I_2 + ·sR_parallel = (R_1 R_2)/(R_1 + R_2) (two resistors)⚠️ Common Mistakes
Misconception: Adding a resistor in parallel increases total resistance.
✓ Correct thinking: Adding any resistor in parallel always DECREASES total resistance, even if the new resistor is large.
Why: Each new parallel path gives current another route to flow, reducing the total opposition. 1/R_total gets larger, so R_total gets smaller.
Misconception: Current is equal through all branches in a parallel circuit.
✓ Correct thinking: Voltage is equal across all branches, but current divides inversely with resistance. Lower resistance → more current.
Why: Each branch sees the same voltage V. By Ohm's law, I = V/R, so branches with smaller R carry more current.
Misconception: The total current in a parallel circuit is found by averaging branch currents.
✓ Correct thinking: Total current is the SUM of all branch currents: I_total = I₁ + I₂ + I₃ + ...
Why: Charge conservation (Kirchhoff's Current Law): all current that leaves the battery must return to it, collecting from all parallel branches.
📝 Practice Problems
Try these problems. Check your answer when ready.
A 6 Ω and 12 Ω resistor are connected in parallel. What is the equivalent resistance?
R_eq = (R_1 R_2)/(R_1 + R_2)Two identical 10 Ω resistors are in parallel. What is the equivalent resistance?
A 4 Ω and 12 Ω resistor are in parallel with a 24 V source. Find the current through each and the total current.
A parallel circuit has two branches. Branch 1 carries 3 A, branch 2 carries 1 A, and the common voltage is 12 V. Find R₁, R₂, and R_total.
Three resistors (2 Ω, 3 Ω, 6 Ω) are in parallel. Find R_eq without using the product-over-sum formula.
(1)/(R_eq) = (1)/(2) + (1)/(3) + (1)/(6)A parallel combination of R₁ and R₂ gives R_eq = 4 Ω. R₁ = 6 Ω. Find R₂.
(1)/(R_eq) = (1)/(R_1) + (1)/(R_2)You have a 12 V battery and need exactly 3 A total current. You only have 6 Ω resistors. How many do you need, and how should they be connected?
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