A bimetallic strip is composed of two identical strips, one of steel and other of brass. When the temperature of the strip is raised by `Delta T` , it bends to form arc of curvature R. If the coefficients of linear expansion of steel and brass are `alpha_1` and `alpha_2` , respectively, then R is

Options

A. directly proportional to `|alpha_1 - alpha_2|`
B. inversely proportional to `|alpha_1 - alpha_2|`
C. directly proportional to `Delta T`
D. NONE OF THESE`

Explanation

To solve the problem of the bimetallic strip composed of steel and brass, we need to analyze how the two materials expand differently when the temperature is raised by ΔT. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the Expansion
When the temperature of the bimetallic strip is raised by ΔT, both materials (steel and brass) will expand. The amount of expansion is determined by their coefficients of linear expansion, denoted as α1 for steel and α2 for brass.

### Step 2: Length Change Due to Expansion
The change in length (ΔL) for each material can be expressed as:
- For brass:
\[ L_b = L_{0b}(1 + \alpha_2 \Delta T) \]
- For steel:
\[ L_s = L_{0s}(1 + \alpha_1 \Delta T) \]

Where \(L_{0b}\) and \(L_{0s}\) are the initial lengths of brass and steel, respectively.

### Step 3: Relating Lengths to Radius of Curvature
When the strip bends, the length of the arc formed by each material can be related to the radius of curvature \(R\) and the angle θ subtended by the arc:
- For brass:
\[ L_b = R \theta \]
- For steel:
\[ L_s = R \theta \]

### Step 4: Setting Up the Equations
Since both materials have the same initial length, we can set up the equations for the lengths after expansion:
1. For brass:
\[ R \theta = L_{0}(1 + \alpha_2 \Delta T) \]
2. For steel:
\[ R \theta = L_{0}(1 + \alpha_1 \Delta T) \]

### Step 5: Dividing the Equations
To find the relationship between R and the coefficients of linear expansion, we can divide the two equations:
\[ \frac{1 + \alpha_2 \Delta T}{1 + \alpha_1 \Delta T} = \frac{R + d}{R} \]
Where \(d\) is the small difference in length due to the different expansions.

### Step 6: Applying the Binomial Approximation
Assuming that the coefficients of linear expansion are small, we can use the binomial approximation:
\[ \frac{1 + \alpha_2 \Delta T}{1 + \alpha_1 \Delta T} \approx 1 + (\alpha_2 - \alpha_1) \Delta T \]

### Step 7: Finding the Radius of Curvature
From the previous steps, we can derive:
\[ \frac{d}{R} \approx \alpha_2 \Delta T - \alpha_1 \Delta T \]
Thus, we can express R as:
\[ R \approx \frac{d}{(\alpha_2 - \alpha_1) \Delta T} \]

### Conclusion
From this derivation, we conclude that the radius of curvature \(R\) is inversely proportional to the difference in the coefficients of linear expansion \(|\alpha_1 - \alpha_2|\) and also inversely proportional to ΔT.

### Final Result
Thus, we can summarize that:
\[ R \propto \frac{1}{|\alpha_2 - \alpha_1| \Delta T} \]