Two conductors have the same resistance at 0° C but their temperature coefficients of resistance are α_1 and α_2. The respective temperature…

Let $R_0$R0​ be the initial resistance of both conductors $\therefore \quad$∴ At temperature $\theta$θ their resistance will be, $R_1=R_0\left(1+\alpha_1 \theta\right)$R1​=R0​(1+α1​θ) and $R_2=R_0\left(1+\alpha_2 \theta\right)$R2​=R0​(1+α2​θ) for, series combination, $\mathrm{R}_{\mathrm{s}}=\mathrm{R}_1+\mathrm{R}_2$Rs​=R1​+R2​ $R_{s 0}\left(1+\alpha_s \theta\right)=R_0\left(1+\alpha_1 \theta\right)+R_0\left(1+\alpha_2 \theta\right)$Rs0​(1+αs​θ)=R0​(1+α1​θ)+R0​(1+α2​θ) where $\mathrm{R}_{\mathrm{s} 0}=\mathrm{R}_0+\mathrm{R}_0=2 \mathrm{R}_0$Rs0​=R0​+R0​=2R0​ $\therefore \quad 2 \mathrm{R}_0\left(1+\alpha_s \theta\right)=2 \mathrm{R}_0+\mathrm{R}_0 \theta\left(\alpha_1+\alpha_2\right)$∴2R0​(1+αs​θ)=2R0​+R0​θ(α1​+α2​) or $\quad \alpha_s=\frac{\alpha_1+\alpha_2}{2}$αs​=2α1​+α2​​ for parallel combination, $$R_p=\frac{R_1 R_2}{R_1+R_2}$$Rp​=R1​+R2​R1​R2​​ where, $R_{p 0}=\frac{R_0 R_0}{R_0+R_0}=\frac{R_0}{2}$Rp0​=R0​+R0​R0​R0​​=2R0​​ $\therefore \quad \frac{\mathrm{R}_0}{2}\left(1+\alpha_{\mathrm{p}} \theta\right)=\frac{\mathrm{R}_0^2\left(1+\alpha_1 \theta+\alpha_2 \theta+\alpha_1 \alpha_2 \theta\right)}{\mathrm{R}_0\left(2+\alpha_1 \theta+\alpha_2 \theta\right)}$∴2R0​​(1+αp​θ)=R0​(2+α1​θ+α2​θ)R02​(1+α1​θ+α2​θ+α1​α2​θ)​ as $\alpha_1$α1​ and $\alpha_2$α2​ are small quantities $\therefore \quad \alpha_1 \alpha_2$∴α1​α2​ is negligible or $\quad \alpha_{\mathrm{p}}=\frac{\alpha_1+\alpha_2}{2+\left(\alpha_1+\alpha_2\right) \theta}=\frac{\alpha_1+\alpha_2}{2}\left[1-\left(\alpha_1+\alpha_2\right) \theta\right]$αp​=2+(α1​+α2​)θα1​+α2​​=2α1​+α2​​[1−(α1​+α2​)θ] as $\left(\alpha_1+\alpha_2\right)^2$(α1​+α2​)2 is negligible $\therefore \quad \alpha_p=\frac{\alpha_1+\alpha_2}{2}$∴αp​=2α1​+α2​​