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. `(a_1+a_2)/2,a_1+a_2`
B. `a_1+a_2,(a_1+a_2)/2`
C. `a_1+a_2,(a_1a_2)/(a_1+a_2)`
D. `(a_1+a_2)/2,(a_1+a_2)/2`

Answer

D

Explanation

Let `R_0` be the initial resistance of both conductors. At temperature their resistances
`R_1=R_0(1+a_1 theta) and R_2=R_0(1+a_2 theta)`
For series combination `R_s=R_1+R_2, R_(s0) (1+a_s theta)=R_0(1+a_1 theta)+R_0(1+a_2 theta)`
where `R_(so)=R_0+R_0=2R_0 therefore 2R_0(1+a_s theta)=2R_0+R_0 theta (a_1+a_2)`
or `a_s=(a_1+a_2)/2` for parallel combination `R_p=(R_1R_2)/(R_1+R_2)`
`R_(p0)(1+a+p theta)=(R_0 (1+a_1 theta) R_0 (1+a_2 theta))/(R_0(1+a_1 theta)+R_0 (1+a_2 theta))`
Where `R_(po)=(R_0R_0)/(R_0+R_0)=R_0/2`
`therefore R_0/2 (1+a_p theta)=(R_0^2(1+a_1 theta+a_2 theta+a_1a_2 theta^2))/(R_0(2+a_1 theta+a_2 theta))`
As `a_1 and a_2` are small quantities
`therefore a_1 a_2` is negligible
or `a_p=(a_1+a_2)/(2+(a_1+a_2) theta) (a_1+a_2)/2[1-((a_1-a_2)/2) theta]`
As `(a_1+a_2)^2` is negligible `therefore a_p=(a_1+a_2)/2`