Two wires of resistances \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) have temperature coefficient of resistances \(\alpha_{1}\) and \(\alpha_{2}\) respectively they are joined in series the effective temperature coefficient of resistance is .... (A) \(\left[\left(\alpha_{1}+\alpha_{2}\right) / 2\right]\) (B) \(\sqrt{\alpha_{1} \alpha_{2}}\) (C) $\left[\left(\alpha_{1} R_{1}+\alpha_{2} R_{2}\right) /\left(R_{1}+R_{2}\right)\right]$ (D) $\left[\sqrt{\left(R_{1}\right.} R_{2} \alpha_{1} \alpha_{2}\right) / \sqrt{ \left.\left(R_{1}^{2}+R_{2}^{2}\right)\right]}$

When two resistors are connected in series, their total resistance, denoted by R_total, can be found using the formula: \[R_{total} = R_1 + R_2\]

The change in resistance for each wire due to a change in temperature can be found using the formula: \[\Delta R = \alpha R \Delta T\] For wire 1, this would be: \[\Delta R_1 = \alpha_1 R_1 \Delta T\] And for wire 2, this would be: \[\Delta R_2 = \alpha_2 R_2 \Delta T\]

Since the wires are connected in series, their resistance changes are added together. So the total change in resistance is the sum of ΔR1 and ΔR2: \[\Delta R_{total} = \Delta R_1 + \Delta R_2\]

Now, substitute the expressions for ΔR1 and ΔR2 from Step 2: \[\Delta R_{total} = \alpha_1 R_1 \Delta T + \alpha_2 R_2 \Delta T\]

We can simplify the expression by factoring out the temperature change ΔT: \[\Delta R_{total} = (\alpha_1 R_1 + \alpha_2 R_2) \Delta T\] Now, comparing with the standard formula for resistance change: \[\Delta R_{total} = \alpha_{total} R_{total} \Delta T\] And substituting the expression for total resistance (Step 1): \[\Delta R_{total} = \alpha_{total} (R_1 + R_2) \Delta T\] Now, we can equate the expressions for ΔR_total obtained in steps 4 and 5: \[(\alpha_1 R_1 + \alpha_2 R_2) \Delta T = \alpha_{total} (R_1 + R_2) \Delta T\] Since ΔT is non-zero, we can divide both sides by ΔT and by R_1 + R_2: \[\alpha_{total} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2}\] This corresponds to answer choice C: \[\left[\left(\alpha_{1} R_{1}+\alpha_{2} R_{2}\right) /\left(R_{1}+R_{2}\right)\right]\]