Ohm's Law
Ohm's law states that, for many conductors at constant temperature, the current through the conductor is proportional to the voltage across it, written V = I × R, where the constant of proportionality R is the resistance. It is the first equation most students meet in electronics, and it ties together three quantities, voltage, current and resistance, so that knowing any two gives the third.
Georg Simon Ohm published the relation in 1827 in a book titled Die galvanische Kette, mathematisch bearbeitet, and it was not warmly received at first. Working as a schoolteacher, Ohm measured how current in wires varied with their length and with the battery used, and cast the results in mathematical form, modelled on Fourier's analysis of heat flow. Many German contemporaries thought a mathematical treatment of electricity too speculative, and recognition came slowly; the Royal Society awarded him its Copley Medal in 1841, and the unit of resistance, the ohm, was later named in his honour. The law's strength was to give circuit behaviour a predictive, quantitative form instead of a collection of qualitative observations.
The worked example shows why the law is so useful, and also how power fits in. Applied to a 5 volt supply across a 1000 ohm resistor, the law gives a current of 5 milliamperes, and the power dissipated, found from P = V × I, is 25 milliwatts. Because substituting Ohm's law gives P = I² × R or P = V² / R, doubling the voltage across a fixed resistor quadruples the heating. The same arithmetic sizes resistors to protect light-emitting diodes, estimates how much a battery will drain, and checks whether a trace on a circuit board will overheat. For networks of resistors, combining Ohm's law with Kirchhoff's rules for currents at junctions and voltages around loops lets any linear circuit be solved.
Ohm's law is a description of certain materials and devices, not a universal law of nature, and the difference matters for transistors. A device is called ohmic if its current-voltage graph is a straight line through the origin, as for metal wires and resistors. A diode is strongly non-ohmic: it conducts almost nothing in reverse and rises exponentially in the forward direction, so no single resistance describes it. A transistor is non-ohmic in a richer way, since its current depends on a control voltage, and in its saturation region the current barely changes with the voltage across it at all. Light bulbs deviate because their filaments heat up, raising resistance. Even for non-ohmic parts one can still define a ratio V / I at any operating point, or a small-signal resistance from the local slope, but this is a convenience and not an instance of the law.
The pitfalls come from treating V = I × R as a causal recipe or a universal truth. Resistance is not a fixed number that the circuit obeys; it is the ratio observed in ohmic conditions. In very thin or very fast situations, such as nanometre wires or high-frequency signals, effects like scattering at boundaries, inductance and capacitance mean the simple law is incomplete, and engineers switch to impedance, the frequency-dependent generalisation. Superconductors, which carry current with zero resistance, are a reminder that the law has limits. Modern chip design relies on Ohm's law constantly nonetheless: voltage drops across power-delivery wiring, termination resistors on high-speed links and the bias networks of analog amplifiers are all first computed with it.
Read Ohm's law as an empirical proportionality that holds for ohmic materials: an extraordinarily useful first approximation for wires and resistors, and a baseline against which the curved, controllable behaviour of diodes and transistors stands out.