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Solve Ohm's Law (V=IR) and power formula (P=VI) for any unknown. Calculate voltage, current, resistance, and power for resistive circuits when two values are known.
Ohm's Law V=IR is the most fundamental relationship in electrical engineering. With the power formula P=VI=I²R=V²/R, four interrelated equations describe any resistive circuit.
| Quantity | Unit | Description | Formula |
|---|---|---|---|
| Voltage (V) | Volts (V) | Electrical potential — pressure | V = IR |
| Current (I) | Amperes (A) | Charge flow rate | I = V/R |
| Resistance (R) | Ohms (Ω) | Opposition to current | R = V/I |
| Power (P) | Watts (W) | Energy per unit time | P = VI = I²R = V²/R |
Series: R_total=R₁+R₂+R₃ (same current). Parallel: 1/R_total=1/R₁+1/R₂ (same voltage). Two 10Ω in series=20Ω. Two 10Ω in parallel=5Ω.
US 120V/15A circuit: max 1800W. At 80% safe load: 1440W. 1500W heater draws 12.5A — near limit! 240V circuits for large appliances.
Ohm's Law (V = IR) is the foundation of electrical circuit analysis. While first published by Georg Simon Ohm in 1827, it remains as relevant today as ever — from analyzing simple resistor circuits to understanding complex electronic systems. Combined with Kirchhoff's laws, it enables analysis of any linear circuit, regardless of complexity.
Kirchhoff's Voltage Law (KVL): the sum of all voltages around any closed loop equals zero. Kirchhoff's Current Law (KCL): the sum of all currents entering a node equals the sum leaving. Combined with Ohm's Law, these three laws allow solution of any linear DC circuit. For a series circuit with battery V and resistors R₁, R₂: V = I(R₁ + R₂), so I = V/(R₁ + R₂). Voltage across R₁: V₁ = IR₁ = V×R₁/(R₁+R₂) — the voltage divider formula. This is the basis of resistor-based sensors, potentiometers, and reference voltage circuits.
Electrical power P = VI = I²R = V²/R. Energy = Power × Time. A 60W light bulb running for 1 hour: Energy = 60W × 3600s = 216,000 J = 0.060 kWh. At $0.12/kWh: cost = $0.0072. Running 24 hours: 1.44 kWh, $0.173. An LED replacement (9W): 0.216 kWh/day, $0.026/day. Over a year: incandescent costs $63, LED costs $9.5 — $53.5 savings per bulb. Understanding power calculations makes energy consumption decisions quantitative.
Ohm's Law applies to linear (ohmic) resistors — materials where V/I ratio is constant. Non-ohmic devices: diodes (current flows preferentially in one direction), transistors (current controlled by a gate voltage), and memristors (resistance depends on history of current flow). Temperature also affects resistance: RTD (resistance temperature detectors) use the predictable increase of metal resistance with temperature for precise thermometry. Thermistors use semiconductors whose resistance decreases sharply with temperature — used in thermometers, fan controls, and battery management systems.
Ohm's Law (V = IR) is the most fundamental relationship in electrical circuit analysis, expressing the linear relationship between voltage (V, electrical potential difference in volts), current (I, charge flow rate in amperes), and resistance (R, opposition to current flow in ohms). Published by Georg Simon Ohm in 1827 from systematic experimental measurements, it remains the starting point for analysis of every resistive electrical circuit, from simple LED circuits to complex power distribution networks.
Combined with Kirchhoff's Voltage Law (sum of voltages around any loop = 0) and Kirchhoff's Current Law (sum of currents at any node = 0), Ohm's Law allows complete analysis of any linear DC circuit. These three laws together are sufficient to solve for all unknown voltages and currents in circuits with any number of components and sources.
Electrical power is the rate of energy transfer: P = VI = I²R = V²/R (all three forms are equivalent by substituting Ohm's Law). A 12V battery connected to a 4Ω resistor: I = V/R = 12/4 = 3A. P = VI = 12 × 3 = 36W. P = I²R = 9 × 4 = 36W. P = V²/R = 144/4 = 36W. Energy consumed: E = P × t. A 60W light bulb running 8 hours/day: E = 60 × 8 = 480 Wh = 0.48 kWh per day. At $0.15/kWh: cost = $0.072/day. Annual: $26.28. An equivalent LED (9W): $3.94/year — $22.34 savings per year per bulb. Scaling to millions of bulbs represents enormous energy savings.
Series circuit (components connected end-to-end): same current flows through all components. Total resistance: R_total = R₁ + R₂ + R₃ + ... Voltage divides proportionally: V_n = I × Rₙ. Voltage divider: V_out = V_in × R₂/(R₁ + R₂). Used in: resistor ladders, reference voltage generators, sensors (thermistors, photoresistors change one resistance in a divider). Parallel circuit (components connected between same two nodes): same voltage across all components. Equivalent resistance: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃. Total resistance is always less than the smallest individual resistance. Current divides inversely with resistance: I_n = V/Rₙ. Most practical circuits combine series and parallel elements.
Resistance depends on material and geometry: R = ρL/A, where ρ = resistivity (Ω·m), L = length (m), A = cross-sectional area (m²). Resistivity values: Silver: 1.59×10⁻⁸ Ω·m (best conductor). Copper: 1.72×10⁻⁸ (standard electrical wiring). Aluminum: 2.82×10⁻⁸ (overhead power lines — lower cost and weight than copper). Iron: 1.00×10⁻⁷. Stainless steel: 6.9×10⁻⁷. Carbon (graphite): 3-60×10⁻⁶. Silicon (semiconductor): 6.4×10⁻⁴ - 2.3×10³ (varies with doping). Glass (insulator): 10¹⁰ - 10¹⁴. PTFE (Teflon): >10¹⁸. Temperature affects resistance: metals increase with temperature (positive temperature coefficient, PTC), semiconductors decrease (negative temperature coefficient, NTC). RTDs (resistance temperature detectors) exploit this for precise thermometry.
Kirchhoff's Voltage Law (KVL): the algebraic sum of all voltage rises and drops around any closed loop equals zero. In a series circuit with battery V_b and resistors R₁, R₂: V_b - IR₁ - IR₂ = 0, so I = V_b/(R₁+R₂). This conservation of energy principle extends to complex multi-loop circuits. Kirchhoff's Current Law (KCL): the algebraic sum of all currents entering and leaving any node equals zero. Conservation of charge: charge cannot accumulate at a node. Together KVL, KCL, and Ohm's Law allow systematic solution of n equations for n unknown voltages/currents in any linear circuit using mesh analysis or node analysis methods.
Ohm's Law applies to linear (ohmic) resistors — where V/I ratio is constant at all voltages and temperatures. Non-ohmic (non-linear) devices: Diodes: conduct current easily in one direction (forward bias, ~0.7V drop for silicon), block in reverse. I-V relationship is exponential, not linear. Transistors: current controlled by a control signal (BJT: base current; MOSFET: gate voltage). Amplification factor β (BJT): I_collector = β × I_base. Memristors: resistance depends on history of current flow — being developed for neuromorphic computing. LED (Light Emitting Diode): non-ohmic, requires current-limiting series resistor to prevent destruction. Design: R_series = (V_supply - V_forward) / I_LED. For 5V supply, LED V_forward = 2V, I_LED = 20mA: R = (5-2)/0.020 = 150Ω.
Electrical safety depends on understanding current and its physiological effects: 1 mA: barely perceptible. 5 mA: safe maximum. 10-20 mA: strong shock, loss of muscle control. 30-50 mA: severe shock, respiratory paralysis. 100 mA: ventricular fibrillation — typically fatal. 1 A: cardiac standstill (may be survivable with defibrillation). Note: it is current, not voltage, that kills — but high voltage can drive lethal current through body resistance (skin: 10,000-100,000 Ω dry, 1,000 Ω wet; internal body: ~500 Ω). At 120V through wet skin: I = 120/1000 = 120 mA — potentially fatal. This is why ground fault circuit interrupters (GFCIs) trip at 5 mA in bathrooms and kitchens. UK/EU 230V systems are inherently more dangerous than US 120V for the same conditions.