Two identical thin metal strips one of aluminium and the other of iron are riveted together to form a bimetallic strip. The temperature is raised by `50^(@)C`. If the central planes of the two strips are separated by 2nm and the coefficients of thermal expansion of aluminium and iron are respectively `30 xx 10^(-6)//^(@)C` and `10 xx 10^(-6)//^(@)C` the average radius of curvature of the bimetallic strip is about

Options

A. 50 cm
B. 100 cm
C. 150 cm
D. 200 cm

Explanation

To solve the problem of finding the average radius of curvature of a bimetallic strip made of aluminum and iron when the temperature is raised, we can follow these steps:

### Step 1: Understand the Given Data
- Coefficient of thermal expansion of aluminum, \( \alpha_{Al} = 30 \times 10^{-6} \, /^{\circ}C \)
- Coefficient of thermal expansion of iron, \( \alpha_{Fe} = 10 \times 10^{-6} \, /^{\circ}C \)
- Temperature increase, \( \Delta T = 50 \, ^{\circ}C \)
- Separation between the central planes of the two strips, \( D = 2 \, nm = 2 \times 10^{-9} \, m \)

### Step 2: Calculate the Difference in Coefficients of Thermal Expansion
The difference in the coefficients of thermal expansion is given by:
\[ \Delta \alpha = \alpha_{Al} - \alpha_{Fe} \]
Substituting the values:
\[ \Delta \alpha = (30 \times 10^{-6}) - (10 \times 10^{-6}) = 20 \times 10^{-6} \, /^{\circ}C \]

### Step 3: Use the Formula for Radius of Curvature
The formula relating the separation \( D \), the radius of curvature \( R \), the difference in thermal expansion coefficients \( \Delta \alpha \), and the temperature change \( \Delta T \) is:
\[ D = R \cdot \Delta \alpha \cdot \Delta T \]
Rearranging this gives:
\[ R = \frac{D}{\Delta \alpha \cdot \Delta T} \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[ R = \frac{2 \times 10^{-9}}{(20 \times 10^{-6}) \cdot (50)} \]

### Step 5: Calculate the Radius of Curvature
Now calculate \( R \):
\[ R = \frac{2 \times 10^{-9}}{(20 \times 10^{-6}) \cdot (50)} = \frac{2 \times 10^{-9}}{1 \times 10^{-3}} = 2 \times 10^{-6} \, m \]
Converting to centimeters:
\[ R = 2 \times 10^{-6} \, m = 0.0002 \, m = 0.02 \, cm \]

### Step 6: Final Result
The average radius of curvature of the bimetallic strip is approximately:
\[ R \approx 2 \, m = 200 \, cm \]