Electricity & CircuitsElectromagnetism 6 min read

Ohm's Law (V = IR) & Kirchhoff's Circuit Laws (KCL & KVL)

How electrical resistance drops voltage along a circuit trace, and how current splits at parallel PCB junctions.

Core FormulaΔV = I · R | ∑ I_in = ∑ I_out | ∑ ΔV_loop = 0

Voltage is electrical push (Volts), Current is charge flow rate (Amps), and Resistance is trace friction (Ohms) that sheds voltage as ΔV = I·R.

Quick Answer in 30 Seconds

Ohm’s Law (V = I·R) states that the voltage drop across any conductor equals the current flowing through it multiplied by its resistance. When a circuit splits into parallel branches, Kirchhoff’s Current Law (∑I_in = ∑I_out) guarantees that total electric charge is conserved—with more current naturally taking the lower-resistance path (I₁/I₂ = R₂/R₁).

Why Microchips Need Dropper Resistors: Sending a 12V battery surge down a 1Ω copper trace causes almost zero voltage drop, frying a 5V or 6V microchip! Switching onto a higher-resistance carbon or nichrome trace safely sheds the excess voltage.
Kirchhoff’s Node Rule (KCL): Charge never vanishes or piles up at a wire junction. Everything flowing in must flow out.
The Wheatstone Bridge Trick: When two parallel resistor arms have matching ratios (R₁/R₂ = R₃/R₄), the voltage difference between their midpoints is exactly 0.0 V—so zero current flows across the bridge!

Playable 60 FPS Circuit Lab

Don't Just Read the Math — Play It

Jump straight into Current Rush: Multi-Rail Circuit Runner in your browser (zero install) and test this exact physical law at 60 FPS:

Play Current Rush (Free)

Part 1 · Ohm’s Law (V = IR)

The High-Speed Copper Trap: Why Zero Resistance Can Fry Your Chip

Every beginner in electronics makes this mistake once: you hook a bright 3.3V or 5V LED directly across a 12V battery using thick copper wire, and—POP!—the LED flashes orange and dies instantly.

Why? Because pure copper has almost zero electrical resistance (R ≈ 0). By Ohm’s Law (ΔV = I · R), passing through near-zero resistance produces near-zero voltage drop. That means the full 12.0 Volts arrives at the delicate semiconductor gate, causing dielectric breakdown and massive overcurrent.

To deliver a safe 6.0 V from a 12.0 V supply, you must route the current through series resistance (like a Nichrome or Carbon trace) so that exactly 6.0 V is dropped along the path (V_socket = V_source - I · R_trace).

Two-panel diagram showing Ohm’s Law voltage drop across copper vs carbon rails on the left, and Kirchhoff’s Current Law parallel branch splitting on the right. — Sketch 6.1: Dropping Voltage with Series Resistance (Panel A) & Splitting Current at a KCL Fork (Panel B)
In Panel A, staying on the 1Ω Copper rail delivers 11.2V (frying the 6V chip), whereas switching to the 10Ω Carbon rail drops voltage right into the 6.0V target window. Panel B shows how an 8A current divides 3:1 across 2Ω and 6Ω parallel branches.
① Low-R Copper (1Ω): Fast drift speed, tiny voltage drop (preserves voltage).② High-R Carbon (10Ω): High electrical friction, rapid voltage drop (sheds excess V).③ KCL Parallel Fork: Lower resistance branch automatically carries proportionally more current.

Interactive 2D Lab · Current Rush: Multi-Rail Circuit Runner

Surf a Live Printed Circuit Board at 60 FPS!

Guide a 12V–24V electron surge across 1Ω Copper, 4Ω Nichrome, and 10Ω Carbon rails. Watch your Live Projected Arrival Voltage and shed just enough potential to power the target IC socket without frying it or browning out!

Try this in-game: In Level 1 (The Copper Sprint), switch between the 1Ω Copper Highway and 10Ω Carbon Track while watching the "EST. END" arrival voltage dial.
Try this in-game: In Level 2 (Kirchhoff’s Dual Fork), hit the KCL Beam Splitter to control two parallel current pulses simultaneously.
Try this in-game: In Level 3 (The Wheatstone Trench), strike the potentiometer triggers to balance R₁/R₂ = R₃/R₄ and null the electric arc.

Play Current Rush (Free)Runs 100% in your browser · No login or install

Part 2 · Kirchhoff’s Laws (KCL & KVL)

How Current Splits at Forks and Balances in Wheatstone Bridges

In 1845, German physicist Gustav Kirchhoff formulated two conservation laws that govern every microchip and power grid on Earth:

1. Kirchhoff’s Current Law (KCL — Conservation of Charge): At any node (junction) where wires meet, the sum of currents entering equals the sum of currents leaving (∑I_in = ∑I_out). When a path splits into two parallel branches R₁ and R₂, both branches share the exact same voltage drop, so their currents split inversely to their resistances: I₁ / I₂ = R₂ / R₁.

2. Kirchhoff’s Voltage Law (KVL — Conservation of Energy): Around any closed circuit loop, the sum of all voltage rises (batteries) minus all voltage drops (resistors) equals zero (∑ΔV_loop = 0).

How a Wheatstone Bridge Creates a "Zero-Current" Safe Zone

Imagine two parallel voltage dividers side by side: Left Arm (R₁ over R₂) and Right Arm (R₃ over R₄), with a detector wire bridging their midpoints. If you tune the resistors so R₁/R₂ = R₃/R₄, the midpoint voltage on the left equals the midpoint voltage on the right (V_left = V_right). With zero voltage difference across the bridge (ΔV = 0), zero current flows through the center—even when powered by a high-voltage source!

Formula Breakdown

How Voltage, Current, and Resistance Interact

Think of a circuit like water flowing through pipes of different widths:

ΔV = I · R and I₁ = I_total · [ R₂ / (R₁ + R₂) ]

Symbol	What It's Called	Plain-English Meaning	How It Feels in Current Rush: Multi-Rail Circuit Runner
`V (or ΔV)`Volts (V)	Potential Difference (Voltage)	Electrical pressure pushing electrons forward; drops by ΔV = I·R across resistive traces.	Must land inside the target IC socket’s rated window (e.g., 6.0 V ± 0.8 V) at the finish line.
`I`Amperes (A)	Electric Current	How many Coulombs of charge pass a point per second (1 Amp = 1 Coulomb/sec).	Spikes on low-resistance copper rails (I = V/R), risking blowing inline fuses if too high!
`R`Ohms (Ω)	Electrical Resistance	Opposition to electron drift caused by atomic lattice collisions.	1Ω Copper preserves voltage and moves fast; 10Ω Carbon drops voltage rapidly.

Try It With Real Numbers

Step-by-Step Math: Sizing a Dropper Resistor & Parallel Current Split

Scenario: Part A: You have a 12.0 V battery and need to deliver exactly 6.0 V to a microcontroller drawing I = 0.60 A. What series trace resistance R is required? Part B: If an 8.0 A surge hits a parallel fork with R₁ = 2.0 Ω and R₂ = 6.0 Ω, how much current flows through each branch?

Step 1 (Part A): Calculate required voltage drop ΔV

ΔV_drop = V_source - V_target = 12.0 V - 6.0 V = 6.0 V

The circuit trace must shed 6.0 V before reaching the microcontroller socket.

Step 2 (Part A): Apply Ohm’s Law to find R

R = ΔV_drop / I = 6.0 V / 0.60 A = 10.0 Ω

A 10.0 Ω carbon dropper trace drops exactly 6.0 V at 0.60 A.

Step 3 (Part B): Apply the KCL Current Divider Formula

I₁ = 8.0 A × [ 6.0 / (2.0 + 6.0) ] = 6.0 A | I₂ = 8.0 A × [ 2.0 / (2.0 + 6.0) ] = 2.0 A

Notice that 6.0 A + 2.0 A = 8.0 A (KCL satisfied), and the 2Ω branch carries 3× more current than the 6Ω branch!

Gameplay Tip: In Current Rush, watch the top-left Voltmeter’s "EST. END" readout: if it predicts you will arrive overvoltage, hop down to the 10Ω Carbon rail to shed ΔV = I·R!

Common Questions

Frequently Asked Questions

Does a resistor slow down the speed of current only after the resistor, or everywhere in a series loop?

Everywhere in the series loop simultaneously! By Kirchhoff’s Current Law, in a single series loop there is nowhere else for electrons to go, so the current (Amps) going into a resistor is identical to the current coming out of it. What drops across the resistor is electrical potential (Voltage), not the number of electrons per second.

Why do household outlets wire appliances in parallel instead of in series?

In a parallel circuit, every branch connects directly across the 120V/240V supply lines, so every appliance gets its full rated voltage and can be switched on or off independently. If your house were wired in series, turning off one bedroom lamp would break the loop and shut off the refrigerator!

What happens in a "short circuit"?

A short circuit occurs when an unintended near-zero resistance path (R → 0 Ω) bridges the power supply terminals. By Ohm’s Law (I = V / R), dividing supply voltage V by a tiny resistance R causes current I to skyrocket to decenas or hundreds of Amps—instantly melting wires or blowing the protective fuse.