Question

Coefficient of linear expansion of brass and steel rods are $\alpha_1$ and $\alpha_2$. Lengths of brass and steel rods are $ l_1$ and $l_2$ respectively. If $(l_2 - l_1)$ is maintained same at all temperatures, which one of the following relations holds good ?

• A. $\alpha_1 l^2_2 = \alpha_2 l_1^2$
• B. $\alpha_1^2 l_2 = \alpha_2^2 l_1$
• C. $\alpha_1 l_1 = \alpha_2 l_2$
• D. $\alpha_1 l_2 = \alpha_2 l_1$

Answer 1

The Correct Option is C

Change in Length for Two Rods 

When two rods are subjected to the same change in temperature, the change in length for both rods should be the same:

\(\Delta \ell_1 = \Delta \ell_2\)

The formula for the change in length is given by the expression:

\(\ell_1 \alpha_1 \Delta T = \ell_2 \alpha_2 \Delta T\)

Since the temperature change \( \Delta T \) is the same for both rods, it cancels out from both sides:

\(\ell_1 \alpha_1 = \ell_2 \alpha_2\)

Conclusion:

This equation indicates that the product of the initial length and the coefficient of linear expansion for each rod must be equal for both rods. Therefore, the relationship between the two rods is:

\(\ell_1 \alpha_1 = \ell_2 \alpha_2\)