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Since heating causes superficial expansion in a conductor, the cross-sectional area increases due to an increase in temperature.

Resistance is inversely proportional to cross-sectional area. So, this implies that resistance should decrease with an increase in temperature in metallic conductors, which is obviously not what is happening.

Where am I wrong in understanding this concept?

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    $\begingroup$ There is more to resistance than cross sectional area. Think about the mechanism. $\endgroup$
    – matt_black
    Commented May 21 at 18:17
  • $\begingroup$ @matt_black I do get the theoretical explanation for this, but I'm not able to make sense of the mathematical implication. $\endgroup$
    – Mel
    Commented May 21 at 18:20
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    $\begingroup$ It is the topic more frequently studied in solid state physics. $\endgroup$
    – Poutnik
    Commented May 21 at 18:35
  • $\begingroup$ Increasing temperatures means increasing atomic motion means increasing electron scattering means increased resistivity. In metals at least… $\endgroup$
    – Jon Custer
    Commented May 21 at 19:08
  • $\begingroup$ For the record you are presumably referring to electrical resistance? $\endgroup$
    – Buck Thorn
    Commented May 22 at 9:50

3 Answers 3

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It depends on the material:

  1. For semiconductors, such as germanium, resistance decreases with an increase in temperature. Semiconductors have a band-gap, and thermal energy promotes the number of conducting electrons and decreases the band gap.
  2. In ordinary conductors such as metals, resistance increases with temperature because electrons are scattered by thermal motion. On a quantum level, one can view it as (thermal) phonon-electron interaction.

Your reasoning about that, "Resistance is inversely proportional to cross-sectional area," is faulty, because the same mass of conductor has not changed, it's just been expanded, increasing distance between atoms. The decrease of resistance with cross-section would only apply if one were to add more material.

BTW, the increase in resistance of metals with temperature can be put to good use. For example, a tungsten incandescent lamp can be put in series with a power supply to make an effective current limiter. At room temperature, the lamp hardly affects current flow, but at its rated current, resistance can increase by a factor of ten or more, preventing excessive current from flowing.

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Resistance does change with temperature. However, it changing due to the cross-sectional area increase is not possible, unless the mass of the conductor changes as well. Understanding how temperature affects a material's resistance isn't as straightforward as it might seem. Two key factors are at play: collisions and the behavior of charge carriers within the material. Generally, as temperature rises, atoms vibrate more, causing more frequent collisions between charge carriers (like electrons) and the atoms themselves. These collisions act like roadblocks, hindering the flow of electricity and increasing resistance. However, some materials, particularly semiconductors, exhibit a negative temperature coefficient. This means their resistance actually decreases with heat. This happens because higher temperatures can generate additional charge carriers within the material. With more carriers available, the overall flow of electricity improves, leading to a decrease in resistance.


The relationship between temperature and resistance often appears linear, especially at room temperature for metals and many other materials. However, this is an oversimplification. At very low temperatures, the collision effect on resistance becomes less important. Here, the mean free path, which is the average distance a charge carrier travels between collisions, takes centre stage. Resistivity (directly related to resistance) is inversely proportional to the mean free path. As temperature increases, the mean free path shrinks, pushing resistance upwards. In essence, for a complete picture, we need to consider both the collision frequency and the behaviour of charge carriers within a specific material when understanding how temperature affects its resistance. Regardless, at temperatures closer to room temperature(25°C), we can understand this by the following equation,

R = Rf(1 + α(T - Tf)

where,

R is the resistance at temperature "T"

Rf is the conductor's resistance at a reference temperature "Tf"

α is temperature coeffecient of resistance for the material

T is the material's temperature

Tf is the reference temperature that α is specified at

*In some cases (T - Tf) can be written as ΔT


Most conductive materials' specific resistance varies with temperature. This is why particular resistance values are typically given at a standard temperature (usually 20° or 25° Celsius).The "α"(temperature coefficient of resistance) is the ratio of resistance change to temperature change per degree Celsius. (There are tabulated values for this)

A positive coefficient for a material indicates that its resistance increases with increasing temperature. Pure metals usually have positive temperature coefficients of resistance. Alloying some metals can result in coefficients that approach zero, however, are still positive. A negative value for a material indicates that its resistance diminishes with increasing temperature. Semiconductor materials (Silicon, Gallium Arsenide, Germanium etc.) usually have negative temperature coefficients of resistance. Insulators also have a negative temperature coefficient.


Sources:

https://www.electronics-notes.com/articles/basic_concepts/resistance/resistance-resistivity-temperature-coefficient.php

https://www.allaboutcircuits.com/textbook/direct-current/chpt-12/temperature-coefficient-resistance/

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Actually, electrical conductance decreases with increase in temperature, while ionic conductance increases with increase in temperature.

For metals, the thermal conductivity is mainly a function of the motion of free electrons. As the temperature increases, the molecular vibrations increase (in turn increasing the mean free path of molecules). So, they obstruct the flow of free electrons, thus reducing the conductivity.

In case of non-metals, there are no free electrons. So, only the molecular vibrations are responsible for conduction of heat and hence for non-metals, the conductivity increases with increase in temperature.

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