We are asked to find the temperature coefficients of resistance for two conductors connected in series and parallel, given their individual temperature coefficients α1\alpha_1 and α2\alpha_2, and that both conductors have the same resistance at 0∘C0^\circ C.
Definitions and Formulae:
1. Temperature Coefficient of Resistance (α\alpha): The temperature coefficient of resistance tells us how much the resistance of a material changes with temperature. It is given by:
RT=R0(1+αΔT)R_T = R_0(1 + \alpha \Delta T)
Where:
o RTR_T is the resistance at temperature TT.
o R0R_0 is the resistance at 0∘C0^\circ C.
o α\alpha is the temperature coefficient of resistance.
o ΔT\Delta T is the change in temperature.
2. For Series Combination: When resistors are in series, the total resistance RtotalR_{\text{total}} is the sum of the individual resistances:
Rtotal=R1+R2R_{\text{total}} = R_1 + R_2
The equivalent temperature coefficient αseries\alpha_{\text{series}} is the average of the individual coefficients, because both resistances increase similarly with temperature:
αseries=α1+α22\alpha_{\text{series}} = \frac{\alpha_1 + \alpha_2}{2}
3. For Parallel Combination: When resistors are in parallel, the total resistance RtotalR_{\text{total}} is given by:
1Rtotal=1R1+1R2\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2}
The equivalent temperature coefficient for parallel resistors is given by the harmonic mean of the individual temperature coefficients:
αparallel=α1α2α1+α2\alpha_{\text{parallel}} = \frac{\alpha_1 \alpha_2}{\alpha_1 + \alpha_2}
Answer:
• The temperature coefficient of the series combination is α1+α22\frac{\alpha_1 + \alpha_2}{2}.
• The temperature coefficient of the parallel combination is α1α2α1+α2\frac{\alpha_1 \alpha_2}{\alpha_1 + \alpha_2}.
Final Answer:
(C) α1+α2,α1α2α1+α2\alpha_1 + \alpha_2, \frac{\alpha_1 \alpha_2}{\alpha_1 + \alpha_2}
