Question

With rise in the temperature, resistance of carbon:
A. Increases
B. Decreases
C. Remains the same
D. First increases then decreases

Answer 1

Hint: We have to find the relation between temperature and resistance in carbon. Since carbon is an insulator, we can find the dependence of resistance and temperature in an insulator. First we can find the change in resistivity with increase in temperature and then by the relation between resistance and resistivity, we can find the change in resistance due to increase in temperature.

Formula used: Resistivity,
$\rho =\dfrac{m}{n{{e}^{2}}\tau }$
Relation between resistance and resistivity,
$\rho =\dfrac{Rl}{A}$

Complete step by step answer:
We have to find the relationship between temperature and resistance in case of carbon.
We know that carbon is an insulator.
Resistivity can be given by the equation,
$\rho =\dfrac{m}{n{{e}^{2}}\tau }$, Where $'\rho '$ is resistivity, $'m'$ is the mass of electron, $'n'$ is number density, i.e. number of free electrons per unit volume, $'e'$ is charge of the electron and $'\tau '$ is average relaxation time.
In the above given equation, we know that ‘m’ and ‘e’, i.e. mass and charge of the electron is constant.
So now let us increase the temperature.
When we increase the temperature ‘n’, the number density increases and$'\tau '$, relaxation time decreases.
The increase in number density is so high, that it compensates for the decrease in relaxation time. Hence the decrease in relaxation time is nullified.
Therefore resistivity is now dependent only on the increase in number density.
From the equation, we can understand that when number density increases, resistivity decreases.
We know that resistivity is directly proportional to resistance.
$\rho =\dfrac{Rl}{A}$
Hence form this we can understand that, when resistivity decreases, resistance also decreases for an insulator.
Since carbon is an insulator, we can conclude that,
When temperature increases, resistance of carbon decreases.

So, the correct answer is “Option B”.

Note: Resistance is the measure of the amount of opposition to the flow of current in an electrical circuit.
Resistivity is the resistance of an object across unit cross sectional area and unit length.
Number density is the number of free electrons per unit volume.
It is calculated by the equation,
$n=\dfrac{N}{V}$ , where ‘N’ is total number of free electrons and ‘V’ is the volume
Relaxation time is the average time gap between two successive collisions.
Mass of an electron$=9.10\times {{10}^{-31}}kg$
Charge of an electron$=1.602\times {{10}^{-19}}C$