Question

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 two metals are ${\alpha }_C$ and ${\alpha }_B.$ On heating, the temperature of the strip increases by $\Delta T$ and the strip bends to form an arc of radius $R$. Then $R$ is proportional to

• A. $\Delta T$
• B. $\frac{1}{\Delta T}$
• C. $\sqrt{\Delta T}$
• D. $\frac{1}{\sqrt{\Delta T}}$

Answer 1

Answer: (B)

Let $L_0$ be the initial length of each strip before heating.
Length after heating will be

$L_B=L_0\left(1+{\alpha }_B\Delta T\right)$
$=(R+d)\theta$
$L_C=L_0\left(1+{\alpha }_C\Delta T\right)=R\theta$
$\Rightarrow \frac{R+d}{R}=\frac{1+{\alpha }_B\Delta T}{1+{\alpha }_C\Delta T}$
$R=\frac{d}{R}=1+\left({\alpha }_B-{\alpha }_C\right)\Delta T$
$R\propto \frac{1}{\Delta T}$