Ohm's Law Calculator

What is Ohm's law?

Electricity flowing through a wire behaves a lot like water flowing through a pipe. Voltage is the push, current is how much actually flows, and resistance is how much the pipe fights back. Ohm's law says these three are locked together.

Turn up the push and more flows. Narrow the pipe, which means more resistance, and less flows. Written as an equation it is about as short as physics gets: voltage equals current times resistance, or V = IR.

Knowing any two of them gives you the third, which is what the calculator above is for. That is genuinely useful. If you want to run a small light off a battery, you need to know what size resistor to put with it so the light gets the current it likes and not more.

There is a fourth number worth having, and that is power, measured in watts. Power tells you how fast energy is being turned into heat and light, and it is just the push multiplied by the flow. It is the reason a kettle needs a thick cable and a doorbell does not.

How do you calculate voltage, current and resistance?

Ohm's law states that the current through a conductor is proportional to the voltage across it, with the constant of proportionality being the resistance:

\[ V = IR, \qquad I = \frac{V}{R}, \qquad R = \frac{V}{I}. \]

Voltage is measured in volts, current in amps and resistance in ohms (Ω). One ohm is the resistance that lets one amp flow under one volt. Electrical power adds a fourth quantity, \(P = VI\), and substituting Ohm's law into it gives the two forms you use when you do not know the current or the voltage directly:

\[ P = VI = I^2R = \frac{V^2}{R}. \]

Those squares matter. Doubling the current through a fixed resistance quadruples the heat, which is why power loss in transmission cables is fought by raising the voltage and lowering the current, not the other way round.

Resistances combine in two patterns. In series, the same current passes through each, so the voltages add and \(R_{\text{total}} = R_1 + R_2 + \dots\). In parallel, each branch sees the same voltage while the current splits, so the reciprocals add: \(1/R_{\text{total}} = 1/R_1 + 1/R_2 + \dots\), and the total is always smaller than the smallest branch. Put those two rules together with Ohm's law and Kirchhoff's laws for junctions and loops, and you can analyse essentially any DC circuit made of resistors.

Ohm's law as a material response, and where it fails

The local form, and what is actually being claimed

The circuit equation \(V = IR\) is a lumped version of a statement about materials: \(\mathbf{J} = \sigma \mathbf{E}\), where \(\mathbf{J}\) is current density, \(\mathbf{E}\) the electric field and \(\sigma\) the conductivity. Integrate that over a uniform wire of length \(L\) and cross-section \(A\) and you recover \(R = \rho L/A\) with \(\rho = 1/\sigma\). Read this way, Ohm's law is not a law of nature in the sense that Maxwell's equations are. It is a constitutive relation, an empirical description of how one particular class of materials responds, and it can be obeyed, bent or broken depending on what the material is doing.

Why a linear response emerges at all

The Drude picture treats conduction electrons as a gas that accelerates in the field and scatters after a mean free time \(\tau\), giving a drift velocity \(v_d = eE\tau/m\) and conductivity \(\sigma = ne^2\tau/m\). The linearity comes from that steady-state balance between acceleration and scattering, not from anything deep. Drude's classical model gets \(\sigma\) roughly right but predicts the wrong electronic heat capacity by two orders of magnitude, and the modern treatment replaces it with Fermi surface transport where only electrons near the Fermi energy carry current. Metals are ohmic because \(\tau\) is essentially independent of the field for ordinary field strengths, and drift velocities are tiny, often under a millimetre per second, even while the signal itself propagates near the speed of light.

Where it stops being true

Plenty of everyday components are deliberately non-ohmic. A diode's current rises exponentially with voltage, following the Shockley equation rather than a straight line. A filament lamp heats up as it conducts, and since \(\rho\) climbs with temperature in metals, its resistance rises with current, so its current-voltage curve bends. Semiconductors move the other way, becoming more conductive when heated, which is why thermal runaway is a real design hazard. At very low temperatures superconductors drop to exactly zero resistance, a state Ohm's law cannot describe at all, and in nanoscale conductors conduction becomes ballistic and quantised in units of \(2e^2/h\), so resistance stops depending on length in the familiar way.

Ohm's own difficulty

When Ohm published in 1827 the result was received badly in Germany, partly because his mathematical treatment sat awkwardly with the prevailing naturphilosophie, and partly because reliable constant-voltage sources barely existed. He had to build his own thermocouple to get a steady source, since the voltaic piles of the day drifted as they discharged. The law was recognised properly in Britain first, and Ohm received the Copley Medal in 1841, more than a decade after the work that now carries his name onto every circuit diagram.