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A plane lamina has an area $2{m^2}$at 10$^ \circ C$ then what is its area at 110$^ \circ C$? It's coefficient of superficial expansion is $2 \times {10^{ - 5}}/C$.
A) 2.002${m^2}$
B) 2.004${m^2}$
C) 2.02${m^2}$
D) 2.04${m^2}$

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Last updated date: 15th Jun 2024
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Answer
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Hint: According to the question we have to name the initial and final area, and find the appropriate temperature by subtraction. Substitute these given values in the formula by some simple steps we can get to the final answer.

Formula used:
$A = {A_0}\left( {1 + \beta \Delta T} \right)$

Complete Step by step solution:
Let A be the final area and and ${A_0}$ be initial area
According to question we have to write given values
A = $2{m^2}$
$\beta $ = $2 \times {10^{ - 5}}/C$

Formula is $A = {A_0}\left( {1 + \beta \Delta T} \right)$
A is the final area at the temperature 110$^ \circ C$
${A_0}$ is the initial area at 10$^ \circ C$
 $\beta $ is the coefficient of superficial expansion
$\Delta T$ is the change in temperature

Calculate change in temperature
$\eqalign{
  & \Rightarrow \Delta T = {T_2} - {T_1} \cr
  & \Rightarrow \Delta T = 110 - 10 \cr
  & \therefore \Delta T = {100^ \circ }C \cr} $

Substitute these values in the formula
$A = {A_0}\left( {1 + \beta \Delta T} \right)$
$\eqalign{
  & \Rightarrow A = 2\left( {1 + 2 \times {{10}^{ - 5}} \times 100} \right) \cr
  & \Rightarrow A = 2\left( {1 + 0.002} \right) \cr
  & \Rightarrow A = 2\left( {1.002} \right) \cr
  & \therefore A = 2.004{m^2} \cr} $

Hence, the correct option is B.

Additional information:

We know that the coefficient of superficial expansion is defined as the increase in area per unit original area at 0$^ \circ C$ per unit rise in temperature.

In the calculation, the magnitude of the coefficient of superficial expansion is so small that it is not necessary to take the initial temperature as 0 °C.

A measure of the warmth or coldness of an object or substance regarding some standard value. The temperature of the two systems is the same when the systems are in thermal equilibrium
Temperature is a physical quantity that expresses hotness and cold and it is measured with a thermometer. The source of the occurrence of heat, It is the manifestation of thermal energy, present in all matter and a flow of energy when a body is in contact with another that is colder.

Note:
We can also solve by taking a little bit of change in the formula, instead of A we use the differential form $\partial A$. It becomes A -${A_0}$ in the same way we can get the final answer. The student must remember we have to find an appropriate temperature.