Solveeit Logo
Login

Question

Question: The temperature coefficient of resistance of conductor varies as \(\alpha \left( T \right)=3{{T}^{2}...

The temperature coefficient of resistance of conductor varies as α(T)=3T2+2T\alpha \left( T \right)=3{{T}^{2}}+2T. If R0R_0 is resistance at T=0 and R is resistance at T, then
A. R=R0(6T+2)R={{R}_{0}}\left( 6T+2 \right)
B. R=2R0(3+2T)R=2{{R}_{0}}\left( 3+2T \right)
C. R=R0(1+T2+T3)R={{R}_{0}}\left( 1+{{T}^{2}}+{{T}^{3}} \right)
D. R=R0(1−T+T2+T3)R={{R}_{0}}\left( 1-T+{{T}^{2}}+{{T}^{3}} \right)

Explanation

Solution

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 α=1R0dRdt\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 R0R_0 is the resistance of a conductor at 0°C and RtR_t is the resistance of the same conductor at t°C, then
Rt=R0(1+αt){{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
α=Rt−R0R0t\alpha =\dfrac{{{R}_{t}}-{{R}_{0}}}{{{R}_{0}}t} or
α=1R0dRdt\alpha =\dfrac{1}{{{R}_{0}}}\dfrac{dR}{dt}
As, Rt−R0t=dRdt\dfrac{{{R}_{t}}-{{R}_{0}}}{t}=\dfrac{dR}{dt} it is the change in resistance.
By rearranging we get,
dR=R0αdTdR={{R}_{0}}\alpha dT
Lets substitute the value given
dR=(3T2+2T)dTdR=\left( 3{{T}^{2}}+2T \right)dT
On integrating both sides,
∫R0RdR=R0∫0T(3T2+2T)dT\int\limits_{{{R}_{0}}}^{R}{dR={{R}_{0}}\int\limits_{0}^{T}{\left( 3{{T}^{2}}+2T \right)}dT}
R−R0=R0(T3+T2)R-{{R}_{0}}={{R}_{0}}\left( {{T}^{3}}+{{T}^{2}} \right)
R=R0(1+T2+T3)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.