Coefficient of linear expnsion of material of resistor is `alpha` . Its temperature coefficient of resistivity and resistance are `alpha_(p)` and `alpha_(R)` respectively , the correct relation is .

Options

A. A. `alpha_(R) = alpha_(rho) - alpha`
B. B. `alpha_(R) = alpha_(rho) + alpha`
C. C. `alpha_(R) = alpha_(rho) + 3 alpha`
D. D. `alpha_(R) = alpha_(rho) - 3alpha`

Answer

To find the correct relation between the coefficient of linear expansion (α), the temperature coefficient of resistivity (α_p), and the temperature coefficient of resistance (α_R), we can follow these steps:

### Step 1: Understand the Definitions
- **Coefficient of Linear Expansion (α)**: This defines how much a material expands per unit length for a unit increase in temperature.
- **Temperature Coefficient of Resistivity (α_p)**: This defines how much the resistivity of a material changes with temperature.
- **Temperature Coefficient of Resistance (α_R)**: This defines how much the resistance of a resistor changes with temperature.

### Step 2: Write the Resistance Formula
The resistance \( R \) of a resistor can be expressed in terms of resistivity \( \rho \), length \( L \), and area \( A \):
\[ R = \frac{\rho L}{A} \]

### Step 3: Express Changes in Length and Area
When the temperature changes by \( \Delta T \):
- The change in length \( L \) can be expressed as:
\[ L = L_0 (1 + \alpha \Delta T) \]
- The change in area \( A \) can be expressed as:
\[ A = A_0 (1 + 2\alpha \Delta T) \]

### Step 4: Substitute Changes into the Resistance Formula
Substituting the expressions for \( L \) and \( A \) into the resistance formula gives:
\[ R = \frac{\rho (L_0 (1 + \alpha \Delta T))}{A_0 (1 + 2\alpha \Delta T)} \]

### Step 5: Express Changes in Resistivity
The change in resistivity \( \rho \) can be expressed as:
\[ \rho = \rho_0 (1 + \alpha_p \Delta T) \]

### Step 6: Substitute Resistivity into the Resistance Formula
Substituting the expression for \( \rho \) into the resistance formula:
\[ R = \frac{\rho_0 (1 + \alpha_p \Delta T) (L_0 (1 + \alpha \Delta T))}{A_0 (1 + 2\alpha \Delta T)} \]

### Step 7: Simplify the Expression
For small changes in temperature, we can ignore higher-order terms:
\[ R \approx R_0 (1 + \alpha_p \Delta T + \alpha \Delta T - 2\alpha \Delta T) \]
This simplifies to:
\[ R \approx R_0 (1 + (\alpha_p - \alpha) \Delta T) \]

### Step 8: Define the Temperature Coefficient of Resistance
The temperature coefficient of resistance \( \alpha_R \) is defined as:
\[ \alpha_R = \frac{1}{R_0} \frac{dR}{dT} \]
From our expression, we can see that:
\[ \alpha_R = \alpha_p - \alpha \]

### Conclusion
Thus, the correct relation is:
\[ \alpha_R = \alpha_p - \alpha \]