Ohm's law and the ohm
Ohm's law says that in many conductors the current is proportional to the voltage, and the factor in that proportion is the resistance: R = U/I. It is an empirical law. It holds on the range where voltage versus current is a straight line through the origin.
The ohm is a derived SI unit: 1 Ω = 1 V/A = 1 kg·m²·s⁻³·A⁻². The definition this calculator uses is the elementary one: the resistance between two points is one ohm when a constant potential difference of one volt produces a current of one ampere. The practical realization of the ohm in metrology uses the quantum Hall effect and the von Klitzing constant. This calculator does not perform that calibration. It divides the two numbers you enter.
History: Georg Simon Ohm, 1827
Georg Simon Ohm (1789–1854) published Die galvanische Kette, mathematisch bearbeitet in Berlin in 1827. He studied galvanic circuits and showed that the current in a given conductor is proportional to the exciting force, which we now write as voltage. Contemporaries received the book coldly. Recognition came later, and Ohm's name became the name of the unit.
In the school form U = I·R the three quantities are peers. This calculator solves the formula for R only. A separate Ohm's law page accepts any two of the three and fills in the third. The split is deliberate: this form has two fields and one unknown, so resistance is not entered as an input.
Model assumptions: linear range, constant temperature, direct current
The model assumes direct current, a temperature that stays fixed during the measurement, and an ohmic element. A diode, a filament that is still heating up, and an electrolyte do not lie on that straight line. Metals increase their resistance with temperature, approximately R = R₀ (1 + α·ΔT). This calculator does not ask for α or for a reference temperature. The result is the resistance at the moment U and I were measured, not a cold catalog value.
On alternating current, impedance takes the place of resistance, and the reactance of a coil or a capacitor depends on frequency. Separate pages compute inductive and capacitive reactance. Here the current is a real number in amperes, and the result is a real number in ohms.
Derivation, conductance, and power from the same pair
Step 1. Measure the voltage drop U across the element and the current I through that element, with consistent arrows. Step 2. Divide: R = U/I, provided the current is not zero. For 12 V and 0.5 A: 12 / 0.5 = 24. The units: a volt divided by an ampere is an ohm. Step 3. Conductance is the reciprocal, G = 1/R = I/U, in siemens. At 12 V and 0.5 A that is exactly 1/24 siemens.
Step 4. The power dissipated in the element at those same values is P = U·I, which is 6 W. The forms P = I²·R and P = U²/R give the same number only when the resistance comes from that same voltage and current. The panel computes power directly as the product, without a second measurement.
Equivalent resistance versus resistivity, series, and parallel
If the element is a uniform wire, resistance is tied to geometry: R = ρ·l/S, where ρ is resistivity, l is length, and S is cross-sectional area. The wire-resistance calculator handles that relation. Resistivity, length, and area do not appear here.
Two resistors in series give a sum of resistances, and two in parallel give a sum of reciprocals. Those combinations live on separate pages. This calculator returns one equivalent resistance for a voltage–current pair, whatever sits between the terminals you measured.
Units, signs, and the limits of the school formula
1 kΩ = 1,000 Ω, 1 MΩ = 1,000,000 Ω, and 1 mΩ = 0.001 Ω. The panel also shows kilohms from 1,000 Ω, megohms from 1,000,000 Ω, and milliohms below 1 Ω. A comma and a period in a field are the same separator. Enter a milliampere as a fraction of an ampere. The unit switch in the header does not change the formula: volts, amperes, and ohms stay.
A short in this model is zero volts at a nonzero current, so R = 0 and conductance is not a finite number. An open circuit, current equal to zero, has no quotient. A negative resistance means the voltage arrow and the current arrow oppose each other. Check the signs on the schematic before you treat the element as a source. The calculator does not reject negative numbers, because the division is then defined.
Sources
G. S. Ohm, Die galvanische Kette, mathematisch bearbeitet, Berlin, 1827. D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, the chapter on current and resistance. H. D. Young and R. A. Freedman, University Physics, the chapter on current and resistance. BIPM, The International System of Units (SI), 9th edition, definition of the ohm. NIST Special Publication 330 and SP 811. This page is for educational purposes.