The electrical resistance of a circuit component or device is defined as the ratio of the voltage applied to the electric current which flows through it:
If the resistance is constant over a considerable range of voltage, then by Ohm's law, I = V/R, can be used to predict the behaviour of the material. Although the definition only involves DC current and voltage, the definition also holds for the AC application of resistors.
Whether or not a material obeys Ohm's law, its resistance can be described in terms of its bulk resistivity. The resistivity is temperature dependent, hence, resistance as well. Over sizeable ranges of temperature, this temperature dependance can be predicted from a temperature coefficient of resistance.
Resistivity[]
The electrical resistance of a wire would be expected to be greater for a longer wire, less for a wire of larger cross sectional area, and would be expected to depend upon the material out of which the wire is made. Experimentally, the dependence upon these properties is a straightforward one for a wide range of conditions, and the resistance of a wire can be expressed as:
where:-
- ρ = resistivity
- L = length of the resistor
- A = cross sectional area of the resistor
The factor in the resistance which takes into account the nature of the material is the resistivity . Although it is temperature dependent, it can be used at a given temperature to calculate the resistance of a wire of given geometry.
It should be noted that it is being presumed that the current is uniform across the cross-section of the wire, which is true only for Direct Current. For Alternating Current there is the phenomenon of "skin effect" in which the current density is maximum at the maximum radius of the wire and drops for smaller radii within the wire. At radio frequencies, this becomes a major factor in design because the outer part of a wire or cable carries most of the current.
The inverse of resistivity is called conductivity. There are contexts where the use of conductivity is more convenient.
Electrical conductivity, σ = 1/ρ
Temperature Coefficient[]
Since the electrical resistance of a conductor such as a copper wire is dependent upon collisional proccesses within the wire, the resistance could be expected to increase with temperature since there will be more collisions, and that is borne out by experiment. An intuitive approach to temperature dependence leads one to expect a fractional change in resistance which is proportional to the temperature change:
where, α is temperature coefficient of resistance
Or, expressed in terms of the resistance at some standard temperature from a reference table:
or
![{\displaystyle R=R_{0}[1+\alpha (T-T_{0})]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/9a5e9d7a7a45bb2567c1014bde38a2e36bdfbfd2)
Low temperature resistivity[]
The temperature dependence of resistivity at temperatures around room temperature is characterized by a linear increase with temperature. Microscopic examination of the conductivity shows it to be proportional to the mean free path between collisions (d), and for temperatures above about 15 K, d is limited by thermal vibrations of the atoms. The general dependence is summarized in the proportionalities:
At extremely low temperatures, the mean free path is dominated by impurities or defects in the material and becomes almost constant with temperature. With sufficient purity, some metals exhibit a transition to a superconducting state.
Resistor combinations[]
The combination rules for any number of resistors in series or parallel can be derived with the use of Ohm's law, the voltage law, and the current law.
Comparison of parallel and series circuits with the same resistors and the same battery voltage applied:-
