Two wires of resistance `R_(1)` and `R_(2)` have temperature coefficient of resistance `alpha_(1)` and `alpha_(2)` respectively. These are joined in series. The effective temperature coefficient of resistance is

Options

A. (a)`(a_(1)+a_(2))/(2)`
B. (b)`sqrt(a_(1)a_(2))`
C. (c)`(a_(1)R_(1)+a_(2)R_(2))/(R_(1)+R_(2))`
D. (d)`(sqrt(R_(1)R_(2)+a_(1)a_(2)))/(R_(1)^(2)+R_(2)^(2))`

Explanation

To find the effective temperature coefficient of resistance for two wires joined in series, we can follow these steps:

### Step 1: Understand the Concept of Temperature Coefficient of Resistance
The temperature coefficient of resistance (α) indicates how much the resistance of a material changes with temperature. It is defined as:
\[ \alpha = \frac{1}{R} \frac{dR}{dT} \]
where \( R \) is the resistance, and \( \frac{dR}{dT} \) is the change in resistance with respect to temperature.

### Step 2: Determine the Equivalent Resistance in Series
When two resistors \( R_1 \) and \( R_2 \) are connected in series, the total or equivalent resistance \( R_{eq} \) is given by:
\[ R_{eq} = R_1 + R_2 \]

### Step 3: Relate the Change in Resistance to Temperature
For each wire, the change in resistance with temperature can be expressed as:
\[ \frac{dR_1}{dT} = \alpha_1 R_1 \quad \text{and} \quad \frac{dR_2}{dT} = \alpha_2 R_2 \]

### Step 4: Write the Equation for the Equivalent Resistance
The change in the equivalent resistance with temperature can be expressed as:
\[ \frac{dR_{eq}}{dT} = \frac{dR_1}{dT} + \frac{dR_2}{dT} \]
Substituting the expressions from Step 3, we get:
\[ \frac{dR_{eq}}{dT} = \alpha_1 R_1 + \alpha_2 R_2 \]

### Step 5: Relate the Effective Temperature Coefficient to the Equivalent Resistance
Now, we can express the effective temperature coefficient \( \alpha_{eq} \) in terms of the equivalent resistance:
\[ \alpha_{eq} R_{eq} = \alpha_1 R_1 + \alpha_2 R_2 \]
Substituting \( R_{eq} = R_1 + R_2 \) into the equation gives:
\[ \alpha_{eq} (R_1 + R_2) = \alpha_1 R_1 + \alpha_2 R_2 \]

### Step 6: Solve for the Effective Temperature Coefficient
To find \( \alpha_{eq} \), we rearrange the equation:
\[ \alpha_{eq} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2} \]

### Final Answer
Thus, the effective temperature coefficient of resistance for the two wires joined in series is:
\[ \alpha_{eq} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2} \]