A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The co-efficients of linear expansion of the two metals are `alpha_(C)` and `alpha_(B)`. On heating, the temperature of the strip goes up by △T the strip bends to form an are of radius of curvature `R`. Then `R` is

Options

A. Proportional to `DeltaT`
B. Inversely proportional to `DeltaT`
C. Proportional to `|alpha_(B)-alpha_(C)|`
D. inversely proportional to `|alpha_(B)-alpha_(C)|`

Explanation

To solve the problem regarding the bimetallic strip formed from copper and brass, we need to derive the expression for the radius of curvature \( R \) when the strip is heated. Here’s a step-by-step solution:

### Step 1: Understanding the Bimetallic Strip
A bimetallic strip consists of two different metals (copper and brass in this case) that expand at different rates when heated. The coefficients of linear expansion are given as \( \alpha_C \) for copper and \( \alpha_B \) for brass.

### Step 2: Coefficients of Linear Expansion
When the temperature of the bimetallic strip increases by \( \Delta T \), the two metals will expand differently due to their different coefficients of linear expansion. This difference in expansion causes the strip to bend.

### Step 3: Deriving the Radius of Curvature
The radius of curvature \( R \) of the bimetallic strip can be derived using the formula:
\[ R = \frac{d}{|\alpha_C - \alpha_B| \cdot \Delta T} \]
where:
- \( d \) is the thickness of the strip,
- \( \alpha_C \) is the coefficient of linear expansion of copper,
- \( \alpha_B \) is the coefficient of linear expansion of brass,
- \( \Delta T \) is the change in temperature.

### Step 4: Analyzing the Expression
From the derived formula, we can see that:
- \( R \) is inversely proportional to \( |\alpha_C - \alpha_B| \) and \( \Delta T \).
- As \( \Delta T \) increases, \( R \) decreases (the strip bends more).
- The absolute difference \( |\alpha_C - \alpha_B| \) also affects the curvature; a larger difference results in a smaller radius of curvature.

### Conclusion
Thus, the final expression for the radius of curvature \( R \) is:
\[ R = \frac{d}{|\alpha_C - \alpha_B| \cdot \Delta T} \]

### Summary of Relationships
1. **Inversely Proportional to \( \Delta T \)**: As temperature increases, the radius of curvature decreases.
2. **Inversely Proportional to \( |\alpha_C - \alpha_B| \)**: A greater difference in expansion coefficients leads to a tighter bend.