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

Answer

c

Explanation

Let `R_(t_1),R_(t_2)` be the resistance of two combination at temperature `t_(1)^(@)C` temperature
Then, `R_(1) =R_(0)(1 + alpha_(1)t)`
and `R_(2) =R_(0)(1 + alpha_(1)t)`
When conduction are in series let `alpha_(S)` be the effective temperature coeffecient of resistances series Then effective resistance at temperature
`t^(@)C` is `R_(1) = R_(t_1) +R_(t_2)`
`2R_(0) (1 + alpha_(S)t)= R_(0)(1+ alpha_(1)t) +(1+ R_(0)(1 + alpha_(2)t)`
or `2 + 2alpha_(1)t = (1+alpha_(1)t ) +(1+alpha_(2)t)`
`= 2 +(alpha_(1)+alpha_(2))t`
On solving `alpha_(s) = (alpha_(1) +alpha_(2))/(2)`
When conductors are in parallel of resistance in parallel. Then `(1)/(R_(P)) = (1)/(R_(t_1)) + (1)/(R_(t_2))`
or `(1)/((R_(0)//2)+[1+alpha_(p)t)) = (1)/(R_(0)(1+alpha_(1)t)) +(1)/(R_(0)(1+alpha_(2)t))`
or `(2)/((1+alpha_(P)t)) = (1)/((1+alpha_(1)t)) + (1)/((1+alpha_(2)t))`
or `2(1+alpha_(p)t)^(-1)=(1+alpha_(1)t)^(-1)+(1+alpha_(2)t)^(-1)`
or `2(1 - alpha_(p)t)0 = ( 1- alpha_(1)t) +(1 - alpha_(2)t)`
On solving `alpha_(p) = (alpha_(1) + alpha_(2))/(2)`