Question

The temperature coefficient of resistance of a conductor varies as $\alpha \left(T\right)=3T^2+2T$. If $R_0$ is resistance at $T=0$ and $R$ is resistance at $T$, then

• A. $R=R_0(6T+2)$
• B. $R=2R_0(3+2T)$
• C. $R=R_0(1+T^2+T^3)$
• D. $R=R_0(1-T+T^2+T^3)$

Answer 1

The correct option is C $R=R_0(1+T^2+T^3)$

Since, the temperature coefficient of a resistance is given by,

$\alpha \left(T\right)=\frac{1}{R_0}\frac{dR}{dT}$

$\Rightarrow (3T^2+2T)=\frac{1}{R_0}\frac{dR}{dT}$

$\Rightarrow dR=R_0(3T^2+2T)dT$

$\Rightarrow {\int }_{R_0}^{R}dR=R_0[3{\int }_0^T{T}^{2}dT+2{\int }_0^{T}TdT]$

$\Rightarrow$ $R=R_0+R_0[T^2+T^3]$

$\Rightarrow$ $R=R_0[1+T^2+T^3]$

Hence, option (c) is the correct answer.