Why materials resist current
As electrons drift through a metal they do not glide freely — they bump into the vibrating atoms of the lattice, losing energy at each collision. This friction-like opposition to current is called resistance. The more a material impedes the flow for a given push, the higher its resistance, and the more energy is shed as heat.
Resistance is the ratio of the voltage across an element to the current through it. The SI unit is the ohm: 1 Ω = 1 volt per ampere.
Ohm's law
For a large class of materials — ordinary metals at fixed temperature — the current is simply proportional to the voltage: double the push, double the flow. That proportionality is Ohm's law, usually written as the relation below.
Ohm's law: voltage equals current times resistance. Rearranged, I = V/R — more voltage or less resistance gives more current.
Resistivity: it's the material, not just the shape
Resistance is a property of a particular object; it depends on both the material and the geometry. A long thin wire resists more than a short fat one of the same metal. To separate the pure material property from the shape, we use resistivity \rho — a number fixed by the substance itself.
Resistance grows with length L and shrinks with cross-sectional area A; ρ (in ohm-metres) captures the material. Longer and thinner means more resistance.
Resistivity spans an enormous range and is exactly what separates a conductor from an insulator. Copper has \rho\approx1.7\times10^{-8}\,\Omega\cdot\text{m} (a superb conductor — that is why wires are copper); glass or rubber sit near 10^{12}\,\Omega\cdot\text{m}, twenty powers of ten higher, which is why they safely sheath the wire.
For metals, resistivity rises roughly linearly with temperature (α > 0): hotter wire, more collisions, more resistance.
Electrical power: where the energy goes
As each charge falls through a potential difference it delivers energy. The rate of that energy delivery is the electrical power. Think of it as (energy per charge) × (charge per second) = voltage × current.
Power delivered to (or by) any circuit element equals current times the voltage across it. The SI unit is the watt: 1 W = 1 J/s.
For an ohmic resistor we can substitute V = IR to get two more forms — pick whichever fits the quantities you know.
In a resistor the electrical energy turns into heat at rate I²R — this is Joule heating (energy dissipation).