A wire of cross-section area `A`, length `L_(1)`, resistivity `rho_(1)` and temperature coefficient of resistivity `alpha_(1)` is connected in series to a second wire of length `L_(2)`, resistivity `rho_(2)` temperature coefficient of resistivity `alpha_(2)` and the same area `A`, so that wires carry same current. Total resistance `R` is independent of temperature for small temperature change if (Thermal expansion effect is negligible)

Options

A. `alpha_(1) = - alpha_(2)`
B. `rho_(1) L_(1) alpha_(1) + rho_(2) L_(2) alpha_(2) = 0`
C. `L_(1) alpha_(1) + L_(2) alpha_(2) = 0`
D. None

Answer

B

Explanation

(b) Let initial resistance of the wires are `R_(1)` and `R_(2)` respectively. Then `R'_(1) + R'_(2) = R_(1) + R_(2)`
`implies R_(1) (1 + alpha_(1) Delta T) + R_(2) (1 + alpha_(2) Delta T)`
`= R_(1) + R_(2)`
`implies R_(1) alpha_(1) + R_(2) alpha_(2) = 0`
`(rho_(1) L_(1))/(A) + (rho_(2) L_(2))/(A) alpha_(2) = 0`
`implies rho_(1) L_(1) alpha_(1) + rho_(2) L_(2) alpha_(2) = 0`