A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The coefficients of linear expansion of the strip goes up by `DeltaT`and the strip bends to from an arc 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)|`

Answer

To solve the problem regarding the radius of curvature \( R \) of a bimetallic strip made of copper and brass when the temperature increases by \( \Delta T \), we will follow these steps:

### Step 1: Understand the Linear Expansion
When the temperature of the bimetallic strip increases, both metals will expand. The linear expansion for each metal can be described by the formula:
\[ L = L_0 (1 + \alpha \Delta T) \]
where \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

### Step 2: Set Up the Equations for Each Metal
For brass (with coefficient of linear expansion \( \alpha_B \)):
\[ L_B = L_0 (1 + \alpha_B \Delta T) \]
For copper (with coefficient of linear expansion \( \alpha_C \)):
\[ L_C = L_0 (1 + \alpha_C \Delta T) \]

### Step 3: Relate Lengths to the Radius of Curvature
When the strip bends to form an arc, the lengths of the brass and copper strips can be expressed in terms of the radius of curvature \( R \) and the angle \( \theta \) subtended at the center:
- Length of brass strip:
\[ L_B = R \theta + d \]
- Length of copper strip:
\[ L_C = R \theta \]
where \( d \) is the additional length due to the bending of the brass strip.

### Step 4: Write the Length Equations
From the linear expansion equations, we have:
1. \( R \theta + d = L_0 (1 + \alpha_B \Delta T) \) (for brass)
2. \( R \theta = L_0 (1 + \alpha_C \Delta T) \) (for copper)

### Step 5: Divide the Two Equations
Dividing the equation for brass by the equation for copper:
\[ \frac{R \theta + d}{R \theta} = \frac{1 + \alpha_B \Delta T}{1 + \alpha_C \Delta T} \]

### Step 6: Simplify the Equation
This simplifies to:
\[ 1 + \frac{d}{R \theta} = \frac{1 + \alpha_B \Delta T}{1 + \alpha_C \Delta T} \]

### Step 7: Use Binomial Approximation
Assuming \( \alpha_B \Delta T \) and \( \alpha_C \Delta T \) are small, we can use the binomial approximation:
\[ \frac{1 + \alpha_B \Delta T}{1 + \alpha_C \Delta T} \approx 1 + (\alpha_B - \alpha_C) \Delta T \]

### Step 8: Rearranging the Equation
From the previous step, we can write:
\[ \frac{d}{R \theta} = (\alpha_B - \alpha_C) \Delta T \]
Thus,
\[ d = R \theta (\alpha_B - \alpha_C) \Delta T \]

### Step 9: Solve for Radius of Curvature \( R \)
Rearranging gives:
\[ R = \frac{d}{\theta (\alpha_B - \alpha_C) \Delta T} \]

### Conclusion
The radius of curvature \( R \) is inversely proportional to the difference in coefficients of linear expansion \( (\alpha_B - \alpha_C) \) and the temperature change \( \Delta T \).

### Final Result
\[ R \propto \frac{1}{(\alpha_B - \alpha_C) \Delta T} \]