Question

A thin copper wire of length $L$ increases in length by 1% when heated from temperature ${T_1}$to ${T_2}$ . What is the percentage change in area when a thin copper plate having dimension $2L \times L$ is heated from ${T_1}$ to ${T_2}$ ?
1%
2%
3%
4%

Answer 1

All the metal when they are heated expands and the factor by which their length changes is called coefficient of linear expansion ($\alpha$) . Since the length of the metal is changing ,its area also increases by a factor called coefficient of areal expansion ($\beta$).

Step by step solution :- initial length of copper wire = $L$
Length after heating from temperature ${T_1}$to ${T_2}$ = $L + (1\% \;of\;L)$=
$(1\% \;of\;L) = L\alpha \Delta T$

$\alpha \Delta T = \dfrac{1}{{100}}$………..(1)
Initial area $A$=$2L \times L$
Area after heating heated from temperature ${T_1}$to ${T_2}$ = $A + \;A\beta \Delta T$
Change in area after heating heated from temperature ${T_1}$to ${T_2}$ =$A$-($A + \;A\beta \Delta T$)
Change in area after heating heated from temperature ${T_1}$to ${T_2}$ = $A\beta \Delta T$
We know that $\beta = 2\alpha$
Putting the value $\beta = 2\alpha$
Percentage % change in area of plate $\dfrac{{A\beta \Delta T}}{A}$=$\beta \Delta T$
$\beta \Delta T = 2\alpha \Delta T$
$\alpha \Delta T = \dfrac{1}{{100}}$
$2\alpha \Delta T = \dfrac{2}{{100}}$= 2%
Hence , the percentage change in area of the copper plate of dimension $2L \times L$is 2%.

Thus option (B) is the correct answer.

Note:- In solid , liquids or gases there is another type of expansion observed when heated from temperature ${T_1}$to ${T_2}$ that is volume expansion. The constant by which the volume changes as we change the temperature from ${T_1}$to ${T_2}$is coefficient of volume expansion $\Upsilon$.
For the isotropic solid materials ( whose lattice structure is well in order ) there is a relation between linear , areal & volumetric coefficients of expansion such that $\alpha :\beta :\Upsilon ::1:2:3$

Thermal expansion of an isotropic object may be imagined as a photographic enlargement.
Liquids usually expand more than solids because the intermolecular forces in liquids are weaker than in solids.

Rubber contracts on heating because in rubber as temperature increases , the amplitude of the transverse vibrations increases more than the amplitude of the longitudinal vibrations.

In cold countries water pipes sometimes burst , because water expands on freezing.