A resistance R carries a current i. The power lost to the surrounding is `lambda (theta - theta_(0))`. Here, `lambda` is a constant, `theta` is temperature of the resistance and `theta_(0)` is the temperature of the atmosphere. If the corfficient of linear expansion is `alpha`. The strain in the resistance is

Options

A. `(alpha)/(lamda) l^(2)R`
B. `alphalambda//R`
C. `(alphal^(2)R)/(2lambda)`
D. proporitonal to the length of the resistance wire

Answer

To find the strain in the resistance \( R \) that carries a current \( i \), we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Power Loss**:
The power lost to the surroundings is given by the equation:
\[ P_{\text{loss}} = \lambda (\theta - \theta_0) \]
where \( \lambda \) is a constant, \( \theta \) is the temperature of the resistance, and \( \theta_0 \) is the temperature of the atmosphere.

2. **Power Generated by the Resistance**:
The power generated due to the current flowing through the resistance is given by:
\[ P_{\text{generated}} = I^2 R \]
At steady state, the power generated equals the power lost:
\[ I^2 R = \lambda (\theta - \theta_0) \]

3. **Rearranging for Temperature Difference**:
From the equation above, we can express the temperature difference \( \theta - \theta_0 \):
\[ \theta - \theta_0 = \frac{I^2 R}{\lambda} \]

4. **Change in Length due to Temperature Change**:
The change in length \( \Delta L \) of the resistance due to the temperature change can be expressed using the coefficient of linear expansion \( \alpha \):
\[ \Delta L = L \alpha \Delta \theta \]
where \( \Delta \theta = \theta - \theta_0 \). Substituting the expression for \( \Delta \theta \):
\[ \Delta L = L \alpha \left(\frac{I^2 R}{\lambda}\right) \]

5. **Calculating Strain**:
The strain \( \epsilon \) in the resistance is defined as the change in length per unit original length:
\[ \epsilon = \frac{\Delta L}{L} \]
Substituting the expression for \( \Delta L \):
\[ \epsilon = \frac{L \alpha \left(\frac{I^2 R}{\lambda}\right)}{L} \]
Simplifying this gives:
\[ \epsilon = \frac{I^2 R \alpha}{\lambda} \]

### Final Answer:
The strain in the resistance is:
\[ \epsilon = \frac{I^2 R \alpha}{\lambda} \]