Ohm's Law

Ohm's law

V = I · R

For an ohmic resistor:

• V — voltage across the resistor (V = volts).
• I — current through it (A = amps).
• R — resistance (Ω = ohms).

Equivalent forms — I = V/R, R = V/I.

Resistance from geometry

R = ρ · L / A

where:

• ρ — resistivity (Ω·m), material property.
• L — length (m).
• A — cross-sectional area (m²).

Material	Resistivity ρ (Ω·m at 20°C)	Type
Silver	1.59 × 10⁻⁸	Best conductor
Copper	1.68 × 10⁻⁸	Most common conductor
Gold	2.44 × 10⁻⁸	—
Aluminum	2.65 × 10⁻⁸	—
Iron	9.71 × 10⁻⁸	—
Mercury	9.6 × 10⁻⁷	Liquid metal
Nichrome	1.10 × 10⁻⁶	Heating elements
Carbon (graphite)	7 × 10⁻⁵	Pencil lead
Silicon (pure)	~640	Semiconductor
Glass	10¹⁰-10¹⁴	Insulator
Quartz	10¹⁶	Excellent insulator
Teflon	10²³-10²⁵	Best insulator

Power in resistor

P = V · I = I² · R = V² / R

All three forms equivalent for ohmic resistor. Watts dissipated as heat.

Real-world resistances

Object	Resistance (Ω)
1 m of 1 mm² copper wire	0.017
1 m of 0.1 mm² copper	0.17
Standard incandescent bulb (60 W, 120 V)	240
Toaster heating element	10-25
Standard 1/4 W resistor (typical values)	10 to 1 MΩ
Human body (dry skin, hand-to-hand)	10⁵ to 10⁶
Human body (wet skin)	1,000 to 10,000
Distilled water	~18 MΩ·cm

JavaScript — Ohm's law calculations

// Basic Ohm's law
function current(V, R) { return V / R; }
function voltage(I, R) { return I * R; }
function resistance(V, I) { return V / I; }
function power(V, I) { return V * I; }

console.log(`9V across 100Ω: ${current(9, 100)} A`);   // 0.09 A
console.log(`Power: ${power(9, 0.09).toFixed(2)} W`);  // 0.81 W

// Resistance from geometry
function wireResistance(rho, length_m, area_m2) {
  return rho * length_m / area_m2;
}

// 1 mm² copper, 10 m
console.log(`10m of 1mm² copper: ${wireResistance(1.68e-8, 10, 1e-6).toFixed(3)} Ω`);
// 0.168 Ω

// Heating element: power dissipation
function heatingElement(voltage, resistance) {
  // P = V²/R
  return voltage * voltage / resistance;
}

// 120 V across 10 Ω
console.log(`Heating element: ${heatingElement(120, 10)} W`); // 1440 W

// Resistance with temperature
function resistanceAtTemp(R_0, alpha, T_celsius) {
  // R(T) = R_0 · (1 + α·ΔT)
  return R_0 * (1 + alpha * (T_celsius - 20));
}

// Copper coil at 100°C
console.log(`Copper +80°C: R = ${resistanceAtTemp(1, 0.00393, 100).toFixed(2)} (× R_0)`);
// 1.314 × — 31% higher

// Voltage divider
function voltageDivider(V_in, R1, R2) {
  return V_in * R2 / (R1 + R2);
}

// Current through parallel branch
function currentInParallel(V, branchR, otherBranchR) {
  // Current flows through branch with R = branchR
  return V / branchR;
}

// Power loss in transmission lines
function transmissionLoss(P_load, V_line, R_line) {
  // I = P/V, then P_loss = I²R
  const I = P_load / V_line;
  return I * I * R_line;
}

// 1000 W at 120 V over 10 Ω wire
console.log(`Loss at 120V: ${transmissionLoss(1000, 120, 10).toFixed(0)} W`);
// 694 W LOST — 70% of power!

// Same load at 12,000 V
console.log(`Loss at 12000V: ${transmissionLoss(1000, 12000, 10).toFixed(2)} W`);
// ~0.07 W — 0.007% (why we use high-voltage transmission)

Where Ohm's law matters

• Circuit design. Selecting resistor values for desired currents and voltages. LED current limiting, voltage dividers, biasing.
• Power transmission. Why we use high voltage (HVDC, AC at 100 kV+) — minimizes I²R losses in long-distance transmission.
• Wire sizing. Ampacity tables (max current for wire gauge) come from R-induced heating limits.
• Heating applications. Toasters, electric stoves, hair dryers, water heaters — controlled by selecting R for desired P.
• Sensors. Strain gauges, thermistors, photoresistors all change R with stimuli — measured via Ohm's law.
• Safety. Body resistance determines shock severity. Higher R = safer (shoes, gloves provide insulation).
• Education. First electrical law taught; foundation for all electrical engineering.

Common mistakes

• Applying it to non-ohmic devices. Diodes, transistors, fluorescent bulbs don't follow V=IR. Use device-specific I-V curves.
• Confusing R and ρ. R is the property of the OBJECT (depends on shape). ρ is the property of MATERIAL (intrinsic). R = ρL/A.
• Ignoring temperature dependence. R changes with T, especially for filaments at incandescence (huge increase).
• Using DC formulas at AC. AC needs impedance Z (frequency-dependent). For pure R, Z = R; for L and C, Z is frequency-dependent.
• Forgetting power dissipation. Resistors heat up; max power rating must be respected. ½ W resistor at 1 W will burn.
• Mixing units. SI uses V, A, Ω. Don't mix mA with Ω directly without converting.