Two wires of resistance `R_(1)` and `R_(2)` have temperature coefficient of resistance `alpha_(1)` and `alpha_(2)` respectively. These are joined in series. The effective temperature coefficient of resistance is

Options

A. `(alpha_1 + alpha_2)/(2)`
B. `sqrt(alpha_1 alpha_2)`
C. `(alpha_1 R_1 + alpha_2 R_2)/(R_1 + R_2)`
D. `(sqrt(R_1 R_2 alpha_1 alpha_2))/(sqrt(R_1^2 + R_2^2))`

Answer

C

Explanation

At temperature `theta` ,
`R'_1 = R_1 (1 + alpha_1 Delta theta), R'_2 = R_2(1 + alpha_2 Delta theta)`
At `0^@C, R_(eq) = R_1 + R_2`
At `theta^@C, R_(eq) = R'_1 + R'_2 = R_1(1 + alpha_1 theta) + R_2 (1 + alpha_2 theta)` …(i)
Let equivalent temperature coefficient is `alpha'`
At `theta^@C, (R_1 + R_2)'_(eq) = (R_1 + R_2) [1 + alpha' theta]` ...(ii)
`(R_1 + R_2) (1 + alpha' theta) = R_1(1 + alpha_1 theta)+ R_2(1 + alpha_2 theta)`
`alpha' = (R_1 alpha_1 + R_2 alpha_2)/(R_1 + R_2)`.