Question

A solid rectangular sheet has two different coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ along its length and breadth respectively. The coefficient of surface expansion is: (for ${\alpha }_1<<1,{\alpha }_2<<1$)

• A. $\frac{{\alpha }_1+{\alpha }_2}{2}$
• B. $2({\alpha }_1+{\alpha }_2)$
• C. $\frac{4{\alpha }_1{\alpha }_2}{{\alpha }_1+{\alpha }_2}$
• D. ${\alpha }_1+{\alpha }_2$

Answer 1

Answer: (D)

${\alpha }_1+{\alpha }_2$

Given:

$\frac{dl}{dT}={\alpha }_{1}l$

$\frac{db}{dT}={\alpha }_{2}b$

Area of the sheet is given by:

$A=lb$

Differentiating wrt temperature, we get:

$\frac{dA}{dT}=l\frac{db}{dT}+b\frac{dl}{dT}$

$\frac{dA}{dT}=lb(\frac{db}{bdT}+\frac{dl}{ldT})$

$\frac{dA}{dT}=A({\alpha }_1+{\alpha }_2)$

Hence, coefficient of surface expansion is given by:

${\alpha }_S={\alpha }_1+{\alpha }_2$