Question

A wire of length 60cm is bent into a circle with a gap of 1cm. At its ends, on heating it by $100°C$, the length of the gap increases to 1.02cm. $\alpha$ of material of wire is
A. ${ 2\times { 10 }^{ -4 } }/{ °C }$
B. ${ 4\times { 10 }^{ -4 } }/{ °C }$
C. ${ 6\times { 10 }^{ -4 } }/{ °C }$
D. ${ 1\times { 10 }^{ -4 } }/{ °C }$

Answer 1

$\alpha$ is the linear coefficient of thermal expansion. Thus, to calculate $\alpha$ use the formula for coefficient of thermal expansion. Substitute the value for $\Delta L$ by taking the difference between final length and initial length. Substitute other values and calculate linear coefficient of thermal expansion

Complete step by step answer:
Given: Initial Length ${L}_{i}$= 1cm
Final length ${L}_{f}$= 1.02cm
Temperature= $100°C$
Formula for linear expansion is given by,
$\Delta L= \alpha L \Delta T$ …(1)
Where, $\Delta L$ is the change in length
$\alpha$ is the linear coefficient of thermal expansion
L is the original length
$\Delta T$ is the change in temperature
Equation. (1) can be written as,
${L}_{f}-{L}_{i}= \alpha {L}_{i} \Delta T$
Substituting the values in above equation we get,
$1.02-1= \alpha \times 1 \times 100$
$\Rightarrow 0.02= \alpha \times 100$
$\Rightarrow \alpha= \dfrac {0.02}{100}$
$\Rightarrow \alpha= { 2\times { 10 }^{ -4 } }/{ °C }$
Thus, the value of linear coefficient of thermal expansion is ${ 2\times { 10 }^{ -4 } }/{ °C }$.
Hence, the correct answer is option A i.e. ${ 2\times { 10 }^{ -4 } }/{ °C }$.

Note:
This expression for linear coefficient of thermal expansion can be used for both when the material is heated as well when the material is cooled. Due to thermal expansion, there is change in area as well. Expression for change in area due to thermal expansion is given by, $\Delta A= 2\alpha A \Delta T$
Where, $\Delta A$ is the change in area of the material.