Ohm's Law

What the law says, in one sentence

Ohm's law is a statement about proportionality: the current flowing through a conductor is proportional to the voltage applied across it. Double the voltage and the current doubles. Halve the voltage and the current halves. Written as an equation, the constant of proportionality is the resistance, and the standard form is:

V = I x R

V is voltage in volts, I is current in amperes, and R is resistance in ohms. The same sentence rearranges into I = V / R for current and R = V / I for resistance, which is where the familiar triangle diagram comes from: cover the quantity you want, and the arrangement of the other two shows the operation.

NASA's Glenn Research Center teaches the law from exactly this starting point, with a 9 V battery feeding an 18 ohm curling iron and the current solved as 9 / 18 = 0.5 A. NASA's classroom activity page adds the wording that matters: current through a conductor is directly proportional to the applied voltage, which is the whole content of the law.

Voltage, current and resistance: the three words

Three words carry the whole law, and mixing them up is the most common source of wrong answers. Voltage is electrical pressure, the push measured in volts. A 9 V battery pushes harder than a 1.5 V cell. Current is the flow, the rate at which charge moves, measured in amperes, always abbreviated A. Resistance is the opposition to that flow, measured in ohms, the Greek letter omega.

The water analogy earns its keep here. Voltage is the pressure in the pipe, current is the gallons per minute coming out, and resistance is how narrow the pipe is. Squeeze the pipe and the same pressure delivers less flow. Swap the pipe for a wider one and the same pressure delivers more. Every rearrangement of the law maps onto the analogy: more pressure with the same pipe means more flow, which is I = V / R.

Why the letter I for current: it comes from the French intensite, the phrase Andre-Marie Ampere used for current intensity, and it stuck long before the SI named the ampere after him. The US Energy Information Administration's lesson on measuring electricity covers the same three definitions with the schoolroom wording.

The three forms, and when each one is handy

One equation, three arrangements, and each one answers a different question. The table shows all three with a single consistent example: a 12 V supply pushing current through a 24 ohm resistor.

The example closes on itself: any two of the three numbers reproduce the third, which is the practical test that a circuit has been measured rather than guessed. It is also why a single multimeter reading is enough. Measure the voltage across a resistor and the current through it, and the resistance falls out of the third row without ever reading the resistance scale.

Where the law came from

Georg Simon Ohm, a German physics teacher, published the relationship in 1827 in a book with the long title Die galvanische Kette, mathematisch bearbeitet, or The Galvanic Circuit Investigated Mathematically. He had measured current through wires of different lengths and thicknesses and found the linear relation between applied voltage and resulting current. The book sank at first, and the Royal Society of London awarded him the Copley Medal in 1841 once the results had been repeated elsewhere.

The unit of resistance, the ohm, was named in his honor when the standardization of electrical units got underway later in the century. Today the ohm is defined through the SI: one ohm is the resistance between two points of a conductor when a constant potential difference of one volt applied between them produces a current of one ampere. That sentence is Ohm's law wearing its formal clothes, and it is what the ohm means rather than an extra rule bolted on.

The power formulas that ride along

Ohm's law alone says nothing about energy, so a companion pair of formulas handles power. Power in watts is voltage times current: P = V x I. Substituting the law into that gives the two forms every electronics course drills: P = I^2 x R and P = V^2 / R.

Those substitutions are why the law matters for safety and not just for grades. Heat in a component grows with the square of current, so doubling the current through the same resistance quadruples the heat. A 6 W LED strip controller and a 1,500 W space heater differ by a factor of 250 in power at the same 120 V outlet, which is why one comes with a power switch and the other with a 15 amp breaker requirement.

Worked check with the NASA example: a 9 V battery across 18 ohms drives I = 9 / 18 = 0.5 A, and the power delivered is P = 9 x 0.5 = 4.5 W. Every one of those numbers can be found with a single multimeter and the three forms in the table above, which is the entire point of the law as a practical tool.

Series, parallel, and combined resistance

The law applies to one component at a time. Real circuits have several, and the equivalent resistance seen by the source decides the total current. Two rules combine resistors into one number that the law can then use.

In series, resistances add: R = R1 + R2 + R3. Two 100 ohm resistors in series look like 200 ohms to the source, and a 12 V supply drives 12 / 200 = 0.06 A through the chain. The same current flows through both, and the supply voltage splits in proportion to the resistances, which is how a voltage divider is built: 12 V across a 100 and a 200 ohm in series leaves 8 V across the larger one, because 200 / (100 + 200) = 0.667 of the total.

In parallel, conductances add, which gives the reciprocal rule: 1 / R = 1 / R1 + 1 / R2. Two 100 ohm resistors in parallel look like 50 ohms, because 1 / (1/100 + 1/100) = 50, and a 12 V supply now drives 12 / 50 = 0.24 A. Parallel always lowers the equivalent resistance below the smallest member, which is why adding outlets to one circuit raises the total current the breaker sees even though each outlet is unchanged.

When the law does not hold

Ohm's law is not a law of nature on the level of gravity. It is an observed property of materials called ohmic, and a short list of everyday parts are not on that list.

Diodes and LEDs. Current rises almost vertically once the forward voltage is reached, so resistance measured at one operating point has no bearing on the next. A series resistor is the fix: it converts a nonlinear part into a circuit whose total behavior is close to linear.

Incandescent filaments. Tungsten at room temperature has a fraction of its hot resistance, so inrush current at switch-on runs several times the steady draw until the filament heats. A bulb that runs 0.5 A glows at 60 W on 120 V in steady state but asks far more of the circuit for its first few milliseconds.

Motors. A stopped motor presents nearly the winding resistance alone, which is small, so locked-rotor current runs multiples of the running current. The running value reflects back EMF as well as resistance, and the law alone will not predict it.

Batteries. A real battery has internal resistance, so terminal voltage drops as load current rises. The law still applies, but to the internal resistance and the load separately, not to the battery as one ideal source.

How this page differs from our Ohm's law calculator

We publish an Ohm's law calculator that takes any two of voltage, current, resistance and power and returns the other two, including heat output and an hourly energy figure. That page is the tool to reach for when you have numbers in hand and want answers out.

This page is the explanation behind it: what the proportionality means, where each of the three forms is the useful one, how series and parallel groupings change the resistance the source sees, and which common parts break the assumption. The two pages pair the way a textbook chapter pairs with its problem set, and the worked examples here, such as the 9 V battery and the divider pair, are computed with the same arithmetic the calculator runs.