Resistance, Resistivity, Conductivity, Insulators, Conductors, and Semiconductors

There is a straight linear relationship between how much current flowed through a material when a given voltage was applied to it. The resistance is the ratio of the applied voltage divided by the resultant current flow. R = V/I is Ohms law where R is the resistance and is abbreviated by the symbol Ω, which is 1V/1A.  Ohm's laws can by applied to ohmic materials, whose resistance remains constant over a variety of voltages. The Resistance is defined by ρ(resistivity), L(length) and A(area). The resistance is ρ x L / A. 

Doubling the cross sectional area will cut the resistance in half, and double the amount of current flowing through the resistor. The current density, denoted by symbol J, represents the rate of current flows per unit area. The drift velocity becomes less as the area increases, and the current density also becomes less as the area increases, since the particles have to travel faster through this area, counting collisions and other various factors. Resistivity also tells how "bad" something is at a passing current, and the conductivity is the inverse of resistivity which is represented by σ. σ = 1/ρ.

V = IR = ρ x (L / A) x I which equates to I x L / σ x A. The conductivity of aluminum is super high, but it produced fire hazards. The conductivity of copper is high as well as gold, silver, iron, while the conductivity of materials that are semiconductors (silicon, copper) are in the middle, whereas the conductivity of sulfur, glass and quartz are among the least conductive.

The resistivity of metals follows the equation

ρ = ρ0[1 + α(T - T0)] where ρ is based on the set of reference resistivity ρ0. α is the temperature coefficient of resistivity, given in units C^-1. 

Water has a very high resistivity, with H3O flowing towards a positive electrode and OH- flowing towards the negative electrode, which results in many ions recombining simultaneously. In contrast, salt which is NaCl, or sodium and chloride, act as charge carriers, and this ions act as charge carriers which lowers the resistance.

So for a current to flow, you must always provide an electric field. Maybe the electrons can go fast enough to collide with air particles, but this takes high amounts of energy. An electron is knocked off an airborne molecule and is attached to Oxygen (O2).

Corona discharge is where ionization is limited to a small region, where the breakdown field strength is exceeded. Spark discharge take place between 2 well rounded conductors, which one of them is grounded. The spark takes place all the way in between the electrodes with the discharge being very fast. This happens in static electricity or spark plugs. Brush discharge can appear as luminescent paths. All discharges from insulators are brush discharges. A vacuum is a near perfect insulator. But there are thermionic emissions and field emissions which are possible, as well as secondary and photoelectric emissions. Without these emissions. there are no charges passing through vacuums. 

The electrical resistivities vary greatly between conductors and insulators. A decent conductor has 10^-8Ωm, and a good insulator has 10^14 Ωm, and a semiconductor has from 10^-5 to 10^3 Ωm depending on temperature. These differences are due to the quantum nature of electrons, which take on many discrete values, coming from the wavelike nature of electrons. When an atom has a lattice, there is more than 1 energy level.

These "gaps" are called energy gaps, which are forbidden regions inside of the electrons, where no travelling wave can exist when placed in the electric potential. These are only possible values of electron energies that may or may not be occupied.

The Pauli exclusion principle has a critical role in determining the properties of materials. It says that no 2 electrons can be in the same quantum state, which has a common divisor of the spin number ms. The electrons further down in energy are the innermost electrons. If we provide an electric field to free electrons, the electrons in lower energy levels cannot accept that energy because they cannot move into a higher level, and the electrons at the top level can accept this region from the electric field. The top of the energy, when it moves freely, there is a current. Semiconductor requires the electron to have to "jump" to make a connection, via a narrow band gap, the energy gap is very small.

A modest electric field will have these electrons to conduct, but there is a minimum electric field to do this. Temperature increases drastically lowers the resistivity. The valence electron leaves a hole and other electrons can move into this hole, leaving behind their own holes, and so forth. Silicon, germanium, and gallium arsenide are all semiconductors. 

A semiconductor with an extra number of valence electron per atom vs silicon, is called an n-type semiconductor, and these electrons are semiconductor. Any atom that has less electrons to form a bond to hold valences together, has holes as charge carriers and have acceptors as impurity atoms. These are called p-type semiconductors, and used in diodes and transistors.