Ohm's Law

V = IR, and the unit algebra that goes with it

Ohm's law is a proportionality with a name for the constant. Put a voltage V across a conductor, measure the current I through it, and for a large class of materials the two turn out to be proportional. Call the constant of proportionality R and you have V = IR, or equivalently I = V/R and R = V/I. Those are not three laws. They are one relation solved for whichever of the three quantities you do not already know, and the practical skill is deciding which one that is before touching the algebra.

Units keep the arithmetic honest. NIST lists the ohm among the SI derived units with special names: it is one volt per ampere, and in base units m^2 kg s^-3 A^-2. The ampere is an SI base unit and the volt is derived, equal to the watt per ampere. So the unit algebra closes on itself, dividing volts by ohms leaves amperes, and 4.30 V divided by 0.0100 A is 430 V/A, which is 430 ohms.

Most errors at this stage are prefix slips rather than conceptual failures. Currents in electronics are usually milliamperes and resistances are usually kilohms, and a misplaced decimal changes an answer by a factor of a thousand. One habit removes most of the risk: notice that volts divided by milliamperes gives kilohms exactly, since 1 V divided by 1 mA is 1000 ohms, which is 1 kilohm. Then 4.30 V divided by 10.0 mA reads straight off as 0.430 kilohm, or 430 ohms, with no exponent bookkeeping at all. Write the unit beside every number you write down, and the units will tell you when you have inverted a ratio.

Ohmic behavior is a finding about a material, not a property of the equation

Georg Ohm published the proportionality in 1827 after experiments on metal wires, and it matters that the result came out of measurement. Despite the name, this is not a law of nature in the way conservation of momentum is. OpenStax puts it in the same category as friction: an experimentally observed regularity that holds for certain materials under certain conditions and is used because it is accurate enough to design with.

The operational test is the current-voltage characteristic. Sweep the applied voltage over a range, positive and negative, and plot the resulting current against it. A component whose characteristic is a straight line through the origin is ohmic, and its resistance is the reciprocal of that line's slope, the same number at every point. A component whose characteristic bends is non-ohmic. Those two sentences are the content of the law. Saying that a component is ohmic is a claim you can falsify with a bench sweep, not an assumption you are entitled to make because the symbol R appears in your equation.

Plenty of components are deliberately non-ohmic, and they are the interesting ones. A silicon diode passes almost nothing when reverse biased and conducts once the forward voltage exceeds roughly 0.7 V, the exact figure depending on the device. Its characteristic is neither straight nor symmetric about the origin, which is precisely what makes rectification possible. A thermistor is a semiconductor resistor, typically a sintered mixture of metal oxides, built so that resistance varies strongly with temperature; NIST describes the common negative-temperature-coefficient type as a nonlinear device whose resistance falls as it warms, which is what makes it a thermometer. An incandescent filament heats itself as it conducts, so its resistance climbs while you are measuring it. In every one of these cases, a single value of R is not so much wrong as meaningless until you say at which operating point it was taken.

The microscopic form: J = sigma E

The device-level statement V = IR mixes two different things together: what the material does, and how much of it there is and in what shape. The microscopic form separates them. Written for a point inside a conductor, Ohm's law says that the current density J, the current per unit cross-sectional area in A/m^2, is proportional to the local electric field E in V/m, with the conductivity sigma as the constant: J = sigma E. The same statement inverted is E = rho J, where rho = 1/sigma is the resistivity in ohm meters. This is closer to how Ohm's result was originally framed, in terms of current density, conductivity and field rather than terminal voltage and total current, and it is the form used in continuum modeling. The tungsten study drawn on later in this lesson writes it exactly that way, as E = rho J, when solving for the current flowing through a heated tungsten tile.

Two consequences are worth carrying forward. First, sigma and rho are properties of the material, so the same copper has the same conductivity in a hair-thin wire and in a bus bar, while R depends on geometry as well; the companion lesson on voltage, current and resistance develops that geometric step. Second, when a material is ohmic, sigma is a constant, so ohmic really means that the conductivity does not depend on the field strength. The moment conductivity starts to depend on field, on temperature, or on which way the current is flowing, the straight line bends, and a single R stops describing the device.

Solving for the unknown: sizing a current-limiting resistor

The routine engineering use of the law is not computing a resistance from a voltage and current you already know. It is placing a resistor so that a current you care about takes the value you want. Suppose a 5.00 V regulated bench supply is to drive a silicon diode at 10.0 mA. The diode is the non-ohmic element and refuses to be summarized by a resistance; over the useful range it simply drops about 0.70 V. Everything else in the loop is ohmic, so the reasoning runs: the resistor must absorb whatever the diode does not, which is 5.00 V minus 0.70 V, or 4.30 V; the same current passes through both because they are in series; therefore R = V/I = 4.30 V divided by 0.0100 A, which is 430 ohms.

That is the whole pattern. Identify which element has its voltage set by physics rather than by you, assign the remaining voltage to the element you control, then solve V = IR for the unknown. It also shows why designers give the resistor a decent share of the supply voltage. Build the circuit with a real 430 ohm resistor at a plus or minus 5 percent tolerance and the current lands between 9.52 mA and 10.53 mA. Assume 0.70 V for a diode that actually drops 0.75 V and the current comes out 9.88 mA instead of 10.00 mA, a shift of 1.2 percent. Had the resistor been allotted only 0.30 V to drop, that same 0.05 V uncertainty in the diode would have moved the current by 16.7 percent. The arithmetic is trivial; deciding how much of the supply to spend on the resistor is the engineering.

Measuring V and I without changing them

Ohm's law is only as good as the two measurements you feed it, and both instruments perturb the circuit they are attached to. An ammeter is placed in series with the component whose current you want, because the current has to pass through it, and it must therefore have very low resistance; OpenStax puts a good one at a fraction of a milliohm. A voltmeter is placed in parallel with the component whose voltage you want, because it must sit across the same two nodes, and it must therefore have very high resistance, quoted at around 10 megohms for an inexpensive instrument and up to 10 gigohms for a precision one.

Put numbers on it. A 5.00 V source drives a 10.0 ohm resistor, so the true current is 0.500 A. Insert an ammeter whose own resistance is 0.50 ohm and the loop becomes 10.5 ohms, so the current falls to 0.476 A, 4.8 percent low, and the meter reports that reduced current perfectly accurately. The instrument is not lying; the circuit is no longer the circuit you meant to measure. Drop the meter resistance to 0.5 milliohm and the error becomes 0.005 percent. Voltmeter loading is the mirror image: a finite meter resistance in parallel pulls the measured node voltage down, and the error matters most when the circuit's own resistances are large compared with the meter's.

Swapping the two instruments is a different order of mistake, and Ohm's law predicts what happens. An ammeter's near-zero resistance placed directly across the 5.00 V source gives 5.00 V divided by 0.50 ohm, which is 10 A, limited only by whatever the source can deliver. A voltmeter placed in series where the ammeter belonged inserts 10 megohms into a loop meant to carry 0.500 A, and the current collapses to 5.00 V divided by 10 000 010 ohms, or 0.500 microampere, a million times less than intended. This lesson is educational material, not guidance for working on electrical installations. Every circuit described here is a low-voltage bench example; real electrical work is governed by codes and carried out by qualified people.

Temperature is the hidden variable

The condition under which a metal behaves ohmically is that its temperature holds still, because resistivity in metals rises as the lattice vibrates harder and scatters the charge carriers more. Over a modest range the effect is captured by a linear coefficient, R = R0 multiplied by (1 + alpha times the temperature change). OpenStax gives alpha as 0.0045 per degree Celsius for tungsten and 0.0039 for copper, and works an example in which a 3.00 kilohm carbon resistor drops to 2.94 kilohms between 20 and 60 degrees Celsius so that the current at 9.00 V rises from 3.00 mA to 3.06 mA, a 2 percent change. Semiconductors move the other way; the same table lists pure silicon at minus 0.075 per degree Celsius, resistivity falling as the material warms and more carriers become available. That sign reversal is the physical basis of the thermistor.

A 2 percent shift is a nuisance you design around. A filament lamp is a different situation entirely, because it operates thousands of degrees above the temperature at which you measured it, and there the linear coefficient model is itself out of its depth. Using recommended fits to critically evaluated experimental data for pure tungsten, the resistivity at 293.15 K is 5.31 microhm centimeters and at 2800 K it is 86.5 microhm centimeters, a ratio of 16.3. The linear coefficient model applied across the same span predicts a ratio of 12.3, roughly 25 percent low. Both numbers say the same thing qualitatively and disagree quantitatively, which is worth knowing before quoting either to three digits: for temperature changes of this size the linear coefficient is an order-of-magnitude tool, not a design value. The worked example carries the consequence through to a current.