Two equal resistance at `0^(@)C` with temperature coefficient of resistance `alpha_(1)` and `alpha_(2)` joined in series act as a single resistance in a circuit the temperature coefficient of their single resistance will be

Options

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

Answer

4

Explanation

At `o^(@)C`, let two resistance be `R_(0)` and `R_(0)`.
At `t^(@)C` let the resistances be `R_(1)` and `R_(2)`.
`therefore R_(1)=R_(0)(1+alpha_(1)t)`
`R_( 2)=R_(0)(1+alpha_(2)t)`
when combined is series.
Temperature coefficient `(beta)=("change in resistance")/("Original resistance"xxt)`
or `beta=("Final resistance"-"Original resistance")/("Original resistance"xxt)`
`beta=((R_(1)+R_(2))-2R_(0))/((2R_(0))t)`
`=R_(0)(1+alpha_(1)t)+R_(0)(1+alpha_(2)t)(-2)/(2R_(0)xxt)`
`=(2+alpha_(1)t+alpha_(2)t-2)/(2t)=(alpha_(1)+alpha_(2))/(2)`