Ohm's Law

Not in the same sense as the conservation of energy. It is better understood as the definition of resistance, R = V/I, together with an empirical observation that for many materials that ratio happens to be constant over a wide range of conditions. Metals, carbon resistors and electrolytes at constant temperature are ohmic to a few parts per million. Semiconductors, diodes, thermistors, filament lamps and gas discharges are emphatically not — their V–I curves bend, kink and saturate. For those devices the resistance is still defined at each operating point as V/I, but it is no longer a fixed number and Ohm's law in its simple form does not predict how the current will change when the voltage changes. In those cases you need the device's actual characteristic curve or a more general model.

Because the electrons moving through the conductor collide with the metal's lattice ions, transferring kinetic energy to them on every collision. The lattice vibrates more vigorously, which is what "hot" means at the microscopic level. The rate at which this energy is transferred is the dissipated power, P = I²R. Because the power depends on the square of the current, doubling the current through a fixed resistance quadruples the heat — which is why a 10 A cable rated for 60 °C will reach well over 200 °C if you pass 20 A through it. This is the physical basis of fuses, circuit breakers and incandescent lamps: every one of them works by deliberately exploiting the I²R heating of a conductor.

Resistivity is a property of the material; resistance is a property of the object. Resistivity ρ is measured in ohm-metres and depends only on what the conductor is made of and its temperature. Copper has ρ = 1.724×10⁻⁸ Ω·m at 20 °C, aluminium 2.65×10⁻⁸, nichrome 1.1×10⁻⁶, and glass around 10¹² — a range of twenty orders of magnitude, which is why copper is used for wires and glass for insulators. Resistance R is what you measure across a specific piece of that material, and it depends on the geometry as well as the material: R = ρL/A. A 1 m length of 1 mm² copper wire has a resistance of 17.2 mΩ, while the same 1 m length of 0.1 mm² wire has 172 mΩ — ten times more, because the cross-section is ten times smaller.

Reduce the network to a single equivalent resistance first, apply Ohm's law to find the total current, then work back to the individual branch currents and voltages. For resistors in series the resistances add directly: R_total = R₁ + R₂ + R₃ + … The same current flows through each, and the voltage divides in proportion to each resistance. For resistors in parallel the reciprocals add: 1/R_total = 1/R₁ + 1/R₂ + … The same voltage appears across each, and the current divides in inverse proportion to each resistance. Two resistors in parallel combine to their product over their sum, R = R₁R₂/(R₁+R₂). Once the equivalent resistance is known, the total current is I = V/R_total, and every other quantity follows from applying Ohm's law locally to each element.

Conventional current flows from the positive terminal of a source, through the external circuit, to the negative terminal. Electron flow — the actual motion of charge carriers in a metal — is in the opposite direction, from the negative terminal through the circuit back to the positive terminal. The convention was established by Benjamin Franklin before the discovery of the electron, and it is retained because it works: all the sign conventions in circuit analysis, from Ohm's law through Kirchhoff's laws to the right-hand rule, assume conventional current. Nothing in circuit theory distinguishes the two, because a moving negative charge in one direction is equivalent to a moving positive charge in the other. Only when you need to think about the physical motion of carriers — in a semiconductor, an electrolyte or a vacuum tube — does it matter which one you are describing.

For purely resistive AC circuits the law applies instantaneously: v(t) = i(t)·R at every moment, so the voltage and current sinusoids are in phase, and their RMS or peak values also obey V = I·R. For circuits containing capacitors and inductors the simple form has to be generalised to V = I·Z, where Z is the impedance — a complex quantity whose magnitude combines resistance and reactance, and whose phase angle tells you how far the current leads or lags the voltage. The magnitude |Z| behaves exactly like a frequency-dependent resistance: |Z| = √(R² + X²), where the reactance X is 1/(ωC) for a capacitor and ωL for an inductor. At low frequencies a capacitor looks like an open circuit and an inductor like a short; at high frequencies they swap roles.

It overheats, and its behaviour becomes progressively less like a resistor. A metal-film resistor running at twice its rated power will reach a surface temperature of 200–300 °C, at which point the value drifts by a few percent, the film oxidises, and the resistance creeps upward. Sustained overload causes the film to crack and the part to fail open — usually silently, but sometimes with a visible scorch mark. Carbon-composition resistors fail by crumbling; wirewound resistors survive much higher overloads because the heat is carried by the wire. The practical rule is to derate: choose a resistor rated for at least twice the power it will actually dissipate, and if it is inside an enclosure or next to other hot components, derate further. Designers of power circuits often use 3× or 5× the nominal rating as margin.