Two conductors have the same resistance at `0^@C` but their temperature coefficient of resistanc are `alpha_1 and alpha_2`. The respective temperature coefficients of their series and parallel combinations are nearly

Options

A. `(alpha_(1)+alpha_(2))/2,alpha_(1)+alpha_(2)`
B. `alpha_(1)+alpha_(2),(alpha_(1)+alpha_(2))/2`
C. `alpha_(1)+alpha_(2),(alpha_(1)alpha_(2))/(alpha_(1)+alpha_(2))`
D. `(alpha_(1)+alpha_(2))/2,(alpha_(1)+alpha_(2))/2`

Answer

D

Explanation

Let `R` be their individual resistance at `0^(@)C`. Their resistance at any other temperature `t` is
`R_(1)=R(1+alpha_(1)t)` and `R_(2)=R(1+alpha_(2)t)`.
In series
`R_("series")=R_(1)+R_(2)=R[2+(alpha_(1)+alpha_(2)).t]`
`=2R[1+(alpha_(1)+alpha_(2)t)/2]`
`alpha_("series")=(alpha_(1)+alpha_(2))/2` In Parallel
`R_("Parallel")=(R_(1)+R_(2))/(R_(1)+R_(2))=(R(1+alpha_(1)t)R(1+alpha_(2)t))/(R(2+alpha_(1)+alpha_(2)t))`
`~~(R^(2)(1+alpha_(1)+alpha_(2))t)/(2R(1+(alpha_(1)+alpha_(2))/2t))
`~~R/2(1+(alpha_(1)+alpha_(2))/2t)`
`alpha_("parallel")=(alpha_(1)+alpha_(2))/2`