Question

The temperature coefficient of resistance of conductor varies as $\alpha \left( T \right)=3{{T}^{2}}+2T$. If $R_0$ is resistance at T=0 and R is resistance at T, then
A. $R={{R}_{0}}\left( 6T+2 \right)$
B. $R=2{{R}_{0}}\left( 3+2T \right)$
C. $R={{R}_{0}}\left( 1+{{T}^{2}}+{{T}^{3}} \right)$
D. $R={{R}_{0}}\left( 1-T+{{T}^{2}}+{{T}^{3}} \right)$

Answer 1

Hint: The temperature coefficient of resistance can be defined as the ratio of increase in resistance per degree rise in temperature to its resistance at 0°C. Its unit is per °C. The expression for temperature coefficient can be given as $\alpha =\dfrac{1}{{{R}_{0}}}\dfrac{dR}{dt}$

Complete step by step answer:
The electrical resistivity of a material is defined as the resistance offered to current flow by a conductor of unit length having a unit area of cross section. The unit is ohm-m.
The resistivity of substances vary with temperature. For conductors, the resistance increases with increases in temperature. If $R_0$ is the resistance of a conductor at 0°C and $R_t$ is the resistance of the same conductor at t°C, then
${{R}_{t}}={{R}_{0}}\left( 1+\alpha t \right)$
Where, α is known as the temperature coefficient of resistance.
The expression for the temperature coefficient can be written as
$\alpha =\dfrac{{{R}_{t}}-{{R}_{0}}}{{{R}_{0}}t}$ or
$\alpha =\dfrac{1}{{{R}_{0}}}\dfrac{dR}{dt}$
As, $\dfrac{{{R}_{t}}-{{R}_{0}}}{t}=\dfrac{dR}{dt}$ it is the change in resistance.
By rearranging we get,
$dR={{R}_{0}}\alpha dT$
Lets substitute the value given
$dR=\left( 3{{T}^{2}}+2T \right)dT$
On integrating both sides,
$\int\limits_{{{R}_{0}}}^{R}{dR={{R}_{0}}\int\limits_{0}^{T}{\left( 3{{T}^{2}}+2T \right)}dT}$
$R-{{R}_{0}}={{R}_{0}}\left( {{T}^{3}}+{{T}^{2}} \right)$
$R={{R}_{0}}\left( 1+{{T}^{2}}+{{T}^{3}} \right)$
Therefore, the correct answer for the given question is option (C).

Note: Metals have positive temperature coefficient, i.e., their resistance increases with increase in temperature. Whereas, insulators and semiconductors have negative temperature coefficient, i.e., their resistance decreases with increase in temperature.