Question

Two wires of resistances R₁ and R₂ have temperature coefficient of resistances α₁ and α₂ respectively they are joined in series the effective temperature coefficient of resistance is ….

(A) [(α₁ + α₂)/2]

(B) √α₁α₂

(C) [(α₁R₁ + α₂R₂)/(R₁ + R₂)]

(D) [√(R₁R₂α₁α₂)/√(R₁² + R₂²)]

Answer 1

The correct option (C) [(α₁R₁ + α₂R₂)/(R₁ + R₂)]

Explanation:

Suppose at t°c, resistance of two wires become R_1t & R_2t & their  equivalent is R_t

Hence Rt = R_1t + R_2t

also  R_1t = R₁(1 + α₁t)

R_2t = R₂(1 + α₂t)

∴ Rt = R₁(1 + α₁t) + R₂(1 + α₂t)

= R₁ + R₂ + (R₁α_­1 + R₂α₂)t

= (R₁ + R₂) [1 + {(R₁α₁ + R₂α₂)/(R₁ + R₂)}t]

∴ effective temp coefficient is {(R₁α₁ + R₂α₂)/(R₁ + R₂)}