Question

Two rods of length $l_1$ and $l_2$ are made of materials whose coefficient of linear expansion are ${\alpha }_1$ and ${\alpha }_2$. If difference between two lengths is independent of temperature then

• A. $\frac{l_1}{l_2}=\frac{{\alpha }_1}{{\alpha }_2}$
• B. $\frac{l_1}{l_2}=\frac{{\alpha }_2}{{\alpha }_1}$
• C. $l_2^2{\alpha }_1=l_1^2{\alpha }_2$
• D. $\frac{{\alpha }_1^2}{l_1}=\frac{{\alpha }_2^2}{l_2}$

Answer 1

The correct option is B $\frac{l_1}{l_2}=\frac{{\alpha }_2}{{\alpha }_1}$

Let's say temperature of both the rods are increased by $\Delta$T then
$l_1^{\prime }=l_1(1+{\alpha }_1\Delta T)l_2^{\prime }=l_2(1+{\alpha }_2\Delta T)l_1^{\prime }-l_2^{\prime }=(l_1-l_2)+(l_1{\alpha }_1-l_2{\alpha }_2)\Delta T\Rightarrow l_1{\alpha }_1=l_2{\alpha }_2$