Thermal Properties Of Matter Question 2
Question 2 - 2024 (31 Jan Shift 1)
Two conductors have the same resistances at $0^{\circ} C$ but their temperature coefficients of resistance are $\alpha _1$ and $\alpha _2$. The respective temperature coefficients for their series and parallel combinations are :
(1) $\alpha _1+\alpha _2, \frac{\alpha _1+\alpha _2}{2}$
(2) $\frac{\alpha _1+\alpha _2}{2}, \frac{\alpha _1+\alpha _2}{2}$
(3) $\alpha _1+\alpha _2, \frac{\alpha _1 \alpha _2}{\alpha _1+\alpha _2}$
(4) $\frac{\alpha _1+\alpha _2}{2}, \alpha _1+\alpha _2$
Show Answer
Answer: (2)
Solution:
Series:
$R _{eq}=R _1+R _2$
$2 R\left(1+\alpha _{eq} \Delta \theta\right)=R\left(1+\alpha _1 \Delta \theta\right)+R\left(1+\alpha _2 \Delta \theta\right)$
$2 R\left(1+\alpha _{\text {eq }} \Delta \theta\right)=2 R+\left(\alpha _1+\alpha _2\right) R \Delta \theta$
Parallel :
$\frac{1}{R _{eq}}=\frac{1}{R _1}+\frac{1}{R _2}$
$\frac{1}{\frac{R}{2}\left(1+\alpha _{eq} \Delta \theta\right)}=\frac{1}{R\left(1+\alpha _1 \Delta \theta\right)}+\frac{1}{R\left(1+\alpha _2 \Delta \theta\right)}$
$2\left[\left(1+\alpha _1 \Delta \theta\right)\left(1+\alpha _2 \Delta \theta\right)\right]$
$=\left[2+\left(\alpha _1+\alpha _2\right) \Delta \theta\right]\left[1+\alpha _{eq} \Delta \theta\right]$
$2\left[1+\alpha _1 \Delta \theta+\alpha _2 \Delta \theta+\alpha _1 \alpha _2 \Delta \theta\right]$
$=2+2 \alpha _{\text {eq }} \Delta \theta+\left(\alpha _1+\alpha _2\right) \Delta \theta+\alpha _{\text {eq }}\left(\alpha _1+\alpha _2\right) \Delta \theta^{2}$
Neglecting small terms
$2+2\left(\alpha _1+\alpha _2\right) \Delta \theta=2+2 \alpha _{\text {eq }} \Delta \theta+\left(\alpha _1+\alpha _2\right) \Delta \theta$
$\left(\alpha _1+\alpha _2\right) \Delta \theta=2 \alpha _{eq} \Delta \theta$
$\alpha _{\text {eq }}=\frac{\alpha _1+\alpha _2}{2}$
