Playground

A simple circuit with flowing charges: adjust voltage and resistance to see current change in real time.

Variables

SymbolNameSIDimensionRange
VVVoltageoutput
Potential difference across the resistor
VM·L²·T⁻³·I⁻¹0 – 240
IICurrent
Electric current through the resistor
AI0 – 20
RRResistance
Electrical resistance of the component
ΩM·L²·T⁻³·I⁻²0.1 – 10000

Deep dive

Derivation
From the microscopic Drude model: electrons accelerate under E, collide with lattice ions with mean free time τ. Drift velocity v_d = eEτ/m_e. Current density J = nev_d = (ne²τ/m_e)E = σE. Integrating over a wire of length L and cross-section A gives V = IR where R = L/(σA).
Experimental verification
Ohm verified using thermocouple voltage sources and a torsion galvanometer. Modern verification spans from microelectronics (MOSFET characterization) to superconducting transitions where R abruptly drops to zero.
Common misconceptions
  • Ohm's law is not universal — it is an empirical law valid only for ohmic materials
  • Resistance is not always constant; it depends on temperature, frequency, and material state
  • Current does not get 'used up' in a resistor — it is the same entering and leaving
Real-world applications
  • Every circuit analysis from phone chargers to power grids
  • Resistance-based temperature sensors (RTDs, thermistors)
  • Current-limiting resistors in LED circuits
  • Fuse and circuit breaker sizing

Worked examples

Current through a resistor

Given:
V:
12
R:
4
Find: I and power P
Solution

I = V/R = 12/4 = 3 A; P = IV = 3 × 12 = 36 W

Voltage divider

Given:
V_in:
9
R1:
3000
R2:
6000
Find: V_out across R₂
Solution

V_out = V_in × R₂/(R₁ + R₂) = 9 × 6000/9000 = 6 V

Scenarios

What if…
  • scenario:
    What if the resistance doubles?
    answer:
    Current halves: I = 12/8 = 1.5 A. Power drops to P = 1.5 × 12 = 18 W — half the original power.
  • scenario:
    What if the wire heats up (resistance increases 20%)?
    answer:
    R becomes 4.8 Ω, I drops to 2.5 A, P = 30 W. Real resistors have temperature coefficients that cause this effect.
  • scenario:
    What if you use a superconductor (R → 0)?
    answer:
    Current becomes limited only by the source's internal resistance. In an ideal circuit, I → ∞ — a short circuit. Superconductors require careful current limiting.
Limiting cases
  • condition:
    R → 0
    result:
    Short circuit (I → ∞ for finite V)
    explanation:
    Zero resistance means unlimited current — a dangerous short circuit in practice.
  • condition:
    R → ∞
    result:
    I → 0 (open circuit)
    explanation:
    Infinite resistance blocks all current flow.
  • condition:
    V → 0
    result:
    I → 0
    explanation:
    No driving voltage means no current through a resistor.

Context

Georg Simon Ohm · 1827

Ohm published his law in 'Die galvanische Kette', initially met with skepticism. It took decades before the physics community recognized the fundamental relationship between voltage, current, and resistance.

Hook

Why does a thin wire heat up more than a thick one carrying the same current?

A 12 V battery drives a current through a 4 Ω resistor. Find the current and power dissipated.

Dimensions: [V] = [I]·[R] → A·Ω = A·(V/A) = V ✓
Validity: Valid for ohmic materials at constant temperature. Non-ohmic devices (diodes, transistors, superconductors) do not obey this law. Resistance itself varies with temperature in real materials.

Related formulas

joule_heatingkirchhoffs_voltage_lawkirchhoffs_current_law