Question

The rods of length L${}_1$ and L${}_2$ are made of materials whose coefficient of linear expansion are ${\alpha }_1$ and ${\alpha }_2$ respectively.If the difference between the two length of 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${}_1^2{\alpha }_1$ =L${}_2^2{\alpha }_2$
• D. ${\alpha }_1^2{L}_1$=${\alpha }_2^2{L}_2$

Answer 1

The correct option is B $\frac{L_1}{L_2}$ = $\frac{{\alpha }_2}{{\alpha }_1}$

On increasing the temperature, the length of both the rods will increase.

If the change in length of both the rods is same irrespective of the rise in temperature, then the difference in the lengths of

two rods can be independent of temperature.

Increase in length of the first rod is,

$\Delta L_1=L_1\alpha \theta$

Increase in length of the second rod is,

$\Delta L_2=L_2\alpha \theta$

The required condition is

$L_1\alpha \theta =L_2{\alpha }_2\theta$

$\Rightarrow \frac{L_1}{L_2}=\frac{{\alpha }_2}{{\alpha }_1}$