Two resistance `R_(1)` and `R_(2)` are made of different material. The temperature coefficient of the material of `R_(1)` is `alpha` and of the material of `R_(2)` is `-beta`. Then resistance of the series combination of `R_(1)` and `R_(2)` will not change with temperature, if `R_(1)//R_(2)` will not change with temperature if `R_(1)//R_(2)` equals

Options

A. `alpha/beta`
B. `(alpha+beta)/(alpha-beta)`
C. `(alpha^(2) + beta^(2))/(alphabeta)`
D. `beta/alpha`

Answer

To solve the problem step by step, we will analyze the resistance of two resistors \( R_1 \) and \( R_2 \) made of different materials, with temperature coefficients \( \alpha \) and \( -\beta \) respectively.

### Step 1: Understand the Temperature Dependence of Resistance
The resistance of a conductor changes with temperature according to the formula:
\[ R_T = R_0 (1 + \alpha T) \]
where \( R_T \) is the resistance at temperature \( T \), \( R_0 \) is the initial resistance, and \( \alpha \) is the temperature coefficient of resistance.

### Step 2: Write the Resistances at Temperature \( T \)
For the resistors \( R_1 \) and \( R_2 \):
- The resistance of \( R_1 \) at temperature \( T \) is:
\[ R_1(T) = R_1 (1 + \alpha T) \]
- The resistance of \( R_2 \) at temperature \( T \) is:
\[ R_2(T) = R_2 (1 - \beta T) \]

### Step 3: Calculate the Total Resistance in Series
The total resistance \( R_s \) of the series combination of \( R_1 \) and \( R_2 \) is given by:
\[ R_s(T) = R_1(T) + R_2(T) = R_1(1 + \alpha T) + R_2(1 - \beta T) \]
Expanding this gives:
\[ R_s(T) = R_1 + R_2 + R_1 \alpha T - R_2 \beta T \]

### Step 4: Set the Condition for Constant Resistance
For the total resistance \( R_s(T) \) to remain constant with temperature, the temperature-dependent terms must cancel out. Therefore, we set:
\[ R_1 \alpha T - R_2 \beta T = 0 \]

### Step 5: Simplify the Equation
Factoring out \( T \) from the equation gives:
\[ T(R_1 \alpha - R_2 \beta) = 0 \]
Since \( T \) cannot be zero for our analysis, we have:
\[ R_1 \alpha = R_2 \beta \]

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]

### Final Answer
Thus, the ratio of the resistances \( R_1 \) and \( R_2 \) is:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]