Conduction is a property of the metallic bond in solids, and it's basically a process that can only be understood using quantum physics. We will discuss it in a crude classical framework.



The "conventional current" points in the same direction as the electric field E in the conductor.

A crude classical approximation to the behavior of conduction electrons in a metallic solid with an electric field present.  Here A is the area of the conductor, n is the number of charge carriers (electrons) per unit volume, e is the magnitude of the electron charge, and vd is the drift speed of the charge carriers.

Georg Ohm (1789 - 1854)
In a conductor, applied voltage and current are proportional. This is not a "law," it's just a property of conductors, and serves as a definition of the concept of resistance. ΔV = I R. To factor out the dependence on the physical dimensions of the conductor, we introduce the conductivity σ. It is also convenient to use a vector quantity to represent current.  This quantity is called the current density, j. It is defined to point in the same direction as E.   Ohm's Rule can then be expressed as E = j/σ = ρj, where the resistivity ρ is defined by ρ = 1/σ.
The power lost in a resistor due to its resistance is P = I2R.



j = E/ρ

As you might expect, for a conductor, the resistance and resistivity increase as the temperature of the conductor increases. The reason is conceptually obvious, since the atoms in a metallic solid vibrate around equilibrium positions, and higher temperature means higher vibrational kinetic energy and frequency... the atoms get in the way of the electrons more often. Over normal ranges of temperature, the dependence is close to linear.  (Room temperature is 300 K!)


Worked Examples.
In a battery, an ongoing chemical reaction produces a constant potential difference between two different conducting plates.



The highest-temperature known superconducting material currently is highly pressurized hydrogen sulfide, the transition temperature of which is 203 K (−70 °C), the highest accepted superconducting critical temperature as of 2015.





A p-n junction “diode” can be biased so that it conducts in only one direction. For details look here.


Doped Semiconductors
Transistors


Transistors replaced vacuum tubes, then tiny silicon chips containing microscopic integrated circuits replaced entire huge boxes of connected electronic components!

Resistors combine in an obvious way. An arrangement that increases the length of the conducting path also increases the resistance, while an arrangement that increases the area of the conducting path also decreases the resistance!


In general we need to keep track of V, R and I, around the circuit.  A voltmeter reads potential difference in Volts, an ammeter reads current in Amps.  We imagine all the resistance in the circuit concentrated at one spot, for convenience.
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