Question

Two wires of resistance ${R_1}$ and ${R_2}$ at ${0^0}C$ have temperature coefficient of resistance ${\alpha _1}$ and ${\alpha _2}$ , respectively. These are joined in series. The effective temperature coefficient of resistance is:
(A) $\dfrac{{{\alpha _1} + {\alpha _2}}}{2}$
(B) $\sqrt {{\alpha _1}{\alpha _2}}$
(C) $\dfrac{{{\alpha _1}{R_1} + {\alpha _2}{R_2}}}{{{R_1} + {R_2}}}$
(D) $\dfrac{{\sqrt {{R_1}{R_2}{\alpha _1}{\alpha _2}} }}{{\sqrt {R_1^2 + R_2^2} }}$

Answer 1

Here, we have the two wires with resistances and their coefficients of resistance at ${0^0}C$ and effective temperature coefficient is asked. So, here we have to use the concept of resistances in series and increase the temperature of each resistance by $t$ .

Complete answer:
Here, we have two resistances and we have to increase their temperature by $t$ , ${\alpha _1}$ and ${\alpha _2}$ are the temperature coefficients of ${R_1}$ and ${R_2}$ , respectively. Let us show it by the diagram below:

Those resistances are given by,
$\begin{gathered} R{t_1} = {R_1}(1 + {\alpha _1}t) \\\ R{t_2} = {R_2}(1 + {\alpha _2}t) \\\ \end{gathered}$ (since, temperature is increased by $t$ )
If these resistances are joined in series then the resistances equivalent is given by
${R_{eq}} = R{t_1} + R{t_2}$
$\Rightarrow {R_{eq}} = {R_1}(1 + {\alpha _1}t) + {R_2}(1 + {\alpha _2}t)$ …..(putting all the values from above)
$= {R_1} + {R_1}{\alpha _1}t + {R_2} + {R_2}{\alpha _2}t$
$= ({R_1} + {R_2}) + t({R_1}{\alpha _1} + {R_2}{\alpha _2})$
$= ({R_1} + {R_2})\left( {1 + \dfrac{{{R_1}{\alpha _1} + {R_2}{\alpha _2}}}{{({R_1} + {R_2})}}t} \right)$
Now, comparing this equation with ${R_{eq}} = R(1 + {\alpha _{eff}}t)$
We observe here that
$R = ({R_1} + {R_2})$ and ${\alpha _{eff}} = \dfrac{{{R_1}{\alpha _1} + {R_2}{\alpha _2}}}{{({R_1} + {R_2})}}$
Here, we obtained the effective temperature coefficient as ${\alpha _{eff}} = \dfrac{{{R_1}{\alpha _1} + {R_2}{\alpha _2}}}{{({R_1} + {R_2})}}$
Thus the correct option is C.

Note:
Here, the wires with different resistances are joined in series and their temperature is increased. Effective temperature coefficient is the resistance-change factor per degree Celsius of temperature.
Both wires have temperature coefficient and hence we obtained the effective temperature coefficient by the above mentioned procedure in the answer.