• Electromagnetism
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  • Elec. resistance and conductance

In an electric circuit, the component which the main purpose is to make it difficult the flow of electrical currents is called electrical resistance or resistor.

Electrical resistance

Important definitions:

Electrical resistance \((R)\):
A resistive element of a circuit is said to be a ohmic resistance if it does not depend on the voltage applied, neither it depends on the direction or intensity of the current (Ohm's Law). For these elements, the electrical resistance is a constant ratio between the potential difference \(\mathbb{V}\) between the conductor terminals and the current intensity \(i\) passing through it, so $$ R = \frac{\mathbb{V}}{i}.$$ The electrical resistance of a solid depends on two factors:
  • The number of free electrons in its structure;
  • The mobility of free electrons through the solid molecules network.
It is the instrument used to measure the electrical resistance.
In the \(IS\) , the unit of electrical resistance is the Ohm, \([R]=\Omega = \frac{V}{A}\).

Ohm's Law

A conductor obeys Ohm's law if the value of its resistance is independent of the potential difference and current \(i\) applied. That is, the potential drop \(\mathbb{V}\), on an ohmic resistance \(R\), which is traversed by a current \(i\), is given by: $$\mathbb{V} = R i.$$ Ohm's law is an empirical law and is valid for some materials. In general, metal conductors are ohmic, but others may not be, as gases or liquids, and other electronic devices such as transistors and diodes. For the latter, the variation of the \(\mathbb{V}\) with the current intensity is not linear. These are called non-ohmic or non-linear conductors. However, we can state that for small variations of \(\mathbb{V}\), almost all conductors of nature obey Ohm 's Law.

Resistivity \((\rho)\)

For a conductor cylinder, made of a material with resistivity \(\rho\) , its electrical resistance is directly proportional to its length \(L\) and inversely proportional to the area \(A\) of its cross-section, such that: $$ R = \rho \frac{L}{A}.$$

The resistivity of a device depends on its geometrical characteristics, in general, its width \(L\) , and the area of its cross section, \(A\).
The unit of resistivity in \(IS\) is Ohm \(\times\) meter, \([\rho]=\Omega m\).

Other quantities related to resistivity are:

Conductance \((G)\)
It is the inverse of the electrical resistance $$G = \frac{1}{R}.$$ The electrical conductance unit is the Siemens, \([G]= S = \frac{1}{\Omega}\).
Conductivity \((\sigma)\)
It is the inverse of resistivity, $$ \sigma = \frac{1}{\rho}.$$

Resistive Components (Resistor)

In schematic electrical circuits, resistors are represented as in the figures above. The first is a constant resistor and the second a variable resistor (rheostat).
The resistor is the electronic component that offers resistance to the passage of electric current, turning electricity into heat. The resistors are widely used:
  • as heat generators (electric irons, ovens);
  • to limiting electrical current;
  • as voltage dividers.
Illustration of a resistor and its color code.
The resistance value and tolerance is given by a color code that is marked on the resistor's surface. The first two colors are the first two digits of the resistance value, the multiplier is a third color and the fourth is tolerance. The value of each color is given by:
Color 1 alg 2nd alg Multip Tolerance
None - - - 20%
Silver - - \(10^{-2}\) 10%
Gold - - \(10^{-1}\) 5%
Black - 0 \(10^{0}\) -
Brown 1 1 \(10^{1}\) 1%
Red 2 2 \(10^{2}\) 2%
Orange 3 3 \(10^{3}\) -
Yellow 4 4 \(10^{4}\) -
Green 5 5 \(10^{5}\) 0.5%
Blue 6 6 \(10^{6}\)
Violet 7 7 \(10^{7}\)
Grey 8 8 \(10^{8}\)
White 9 9 \(10^{9}\)

In addition to the resistor of constant resistance, there are resistors with variable resistance, known as rheostats. There are rheostats that can change continuously between certain resistance limits, and there are others whose resistance can only assume discrete values.

Resistance Variation with Temperature

For a conductor to obey Ohm's law will depend on how its resistance changes with temperature. Once there is an increase in temperature the vibration of the molecules of the material increases, and the number of collisions will greater between the moving charges and vibrating molecules. With this, resistance increases since the current has more difficulty to flow.

A resistive material \(R_0\) at a temperature \(T_0\) have a higher resistance \(R\) when undergoing a temperature change \(T - T_0\) . For many materials, the relationship between temperature and resistance is given by the equation $$R=R_0[1+\alpha(T-T_0)],$$ where \(\alpha\) is the coefficient of the thermal variation of the conductor resistance.



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