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

Answer

A

Explanation

`R_(1) = R_(0) (1 + alpha_(1)t)`
`R_(2) = R_(0)(1 + alpha_(2)t)`
In series,
`R = R_(1) + R_(2) = R_(0)[2 + (alpha_(1) + alpha_(2))t]`
`= 2R_(0)[1 + (alpha_(1) + alpha_(2))/2t]`
`therefore alpha_(s) = (alpha_(1) + alpha_(2))/2`
In parallel,
`1/R= 1/R_(1) + 1/R_(2)`
`rArr 1/(R_(0)/2(1+alphat)) =1/(R_(0)(1 + alpha_(1)t)) +1/(R_(0)(1 + alpha_(2)t))`
Using binomial expansion,
`rArr 2/R_(0)[1-alphat]=1/R_(0)[1-alpha_(1)t + 1-alpha_(2)t]`
`rArr 2-2alphat =2-[alpha_(1) + alpha_(2)]t`
`alpha =(alpha_(1) + alpha_(2))/2`