Two conductors have the same resistance at \(0^{\circ} \mathrm{C}\) but their temperature coefficients of resistances are \(\alpha_{1}\) and \(\alpha_{2}\) The respective temperature coefficients of their series and parallel combinations are nearly.... (A) $\alpha_{1}+\alpha_{2},\left\\{\left(\alpha_{1} \alpha_{2}\right) /\left(\alpha_{1}+\alpha_{2}\right)\right\\}$ (B) $\left\\{\left(\alpha_{1}+\alpha_{2}\right) / 2\right\\},\left\\{\left(\alpha_{1}+\alpha_{2}\right) / 2\right\\}$ (C) $\left\\{\left(\alpha_{1}+\alpha_{2}\right) / 2\right\\}, \alpha_{1}+a_{2}$ (D) \(\alpha_{1}+\alpha_{2},\left\\{\left(\alpha_{1}+a_{2}\right) / 2\right\\}\)

First, we start by recalling the formula used to find the resistance of a conductor at any given temperature: \[R_T = R_0(1 + \alpha T)\] Where \(R_T\) represents the resistance at temperature T, \(R_0\) represents the initial resistance (at 0°C), \(\alpha\) is the temperature coefficient of resistance, and T is the temperature in Celsius.

Now, let's recall the formula for equivalent resistance in a series and parallel combination: For a series combination, the total resistance is the sum of individual resistances: \[R_{series} = R_{1} + R_{2}\] For a parallel combination, the reciprocal of the total resistance is equal to the sum of the reciprocals of individual resistances: \[\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2}\]

Now, since two conductors have the same resistance at 0°C, we write: \[R_0 = R_{1_0} = R_{2_0}\] Now let's analyze the series combination: \[ R_{series} = R_{1_T} + R_{2_T} = R_0(1 + \alpha_{1} T) + R_0(1 + \alpha_{2} T) = R_0(2 + (\alpha_{1}+\alpha_{2}) T) \] As seen in the series combination formula above, the temperature coefficient of the series combination is: \[\alpha_{series} = \alpha_{1} + \alpha_{2}\] Now, let's analyze the parallel combination: \[ \frac{1}{R_{parallel}} = \frac{1}{R_{1_T}} + \frac{1}{R_{2_T}} = \frac{1}{R_0(1 + \alpha_{1} T)} + \frac{1}{R_0(1 + \alpha_{2} T)} \] By multiplying and dividing the numerator and denominator by \((1 + \alpha_{1} T)(1 + \alpha_{2} T)\) we get: \[ \frac{1}{R_{parallel}} = \frac{(1 + \alpha_{1} T) + (1 + \alpha_{2} T)}{R_0(1 + \alpha_{1} T)(1 + \alpha_{2} T)} \] \[ R_{parallel} = R_0 \frac{(1 + \alpha_{1} T)(1 + \alpha_{2} T)}{(1 + \alpha_{1} T) + (1 + \alpha_{2} T)} \] Since \(\alpha_{1}T\) and \(\alpha_{2}T\) are small in comparison to 1, we can simplify the expression as: \[R_{parallel} = R_0 \left(1 + \frac{\alpha_{1}\alpha_{2} T^{2}}{(\alpha_{1}+\alpha_{2}) T}\right) \] The temperature coefficient of the parallel combination would therefore approximately be: \[\alpha_{parallel} = \frac{ \alpha_{1} \alpha_{2}}{\alpha_{1}+\alpha_{2}}\]

So, the respective temperature coefficients of their series and parallel combinations are nearly: (A) \(\alpha_{1}+\alpha_{2},\left\\{\left(\alpha_{1} \alpha_{2}\right) /\left(\alpha_{1}+\alpha_{2}\right)\right\\}\)