Question

A metal sheet having size of $0.6 \times 0.5 m$ is heated from \[293{\text{ }}K\] to $520^\circ C$. The final area of the hot sheet is $\left\{ {{\alpha_{metal}} = 2 \times {{10}^{ - 5}}/^\circ C} \right\}$
A. \[0.306{m^2}\]
B. \[0.0306{m^2}\]
C. \[3.06{m^2}\]
D. \[1.02{m^2}\]

Answer 1

Hint: As in the given question, initial and final temperature is given. We need to calculate the area of the hot sheet. When the temperature increases then the area will also increase.
By using the given formula, we can calculate the area of the metal sheet.
${A_2} = A(1 + \beta \times ({t_2} - {t_1}))$
Here $A_2$ is the final area and ($t_2-t_1$) is the change in temperature, \[\beta \] is the thermal expansion coefficient.

Complete step by step solution:
Given: Initial temperature ${t_1} = 293K$
Final temperature ${t_2} = $$520^\circ C$ = \[793{\text{ }}K\]
Length of metal sheet = \[0.6{\text{ }}m\]
Breadth of metal sheet = \[0.5{\text{ }}m\]
${\alpha _{metal}} = 2 \times {10^{ - 5}}/^\circ C$
Metals are good conductors of heat and electricity. When heat is given to any metal, the particles of metals start vibrating to conduct electricity from hot area to cold area. These atomic vibrations cause expansion in the area of metal or thermal expansion. Since we know,
$\beta = 2 \times \alpha $
We can calculate $\beta $
$\therefore \beta = 2 \times 2 \times {10^{ - 5}}$
$\therefore \beta = 4 \times {10^{ - 5}}$ ...............………………………… (I)
Now, the difference between the temperatures is given by subtracting initial temperature from final temperature.
$\therefore {t_2} - {t_1} = 793 - 293 = 500$ …………………………………..(II)
Now area of metal sheet initially (A) = \[Length{\text{ }}of{\text{ }}metal{\text{ }}sheet{\text{ }} \times {\text{ }}Breadth{\text{ }}of{\text{ }}metal{\text{ }}sheet\]
area of metal sheet initially (A) = \[0.6{\text{ }} \times {\text{ }}0.5\]
area of metal sheet initially(A) = $0.3{m^2}$ …………….(III)
When the temperature is increased the length, breadth, area and volume of a metal, is automatically increased. This process is called thermal expansion.

Area of hot sheet when temperature is increased =${A_2} = A(1 + \beta \times ({t_2} - {t_1}))$……………….(IV)
Put values of equation (I), (II) and (III) in equation (IV):
$\Rightarrow {A_2} = 0.3 \times (1 + (4 \times {10^{ - 5}}) \times 500)$
$\Rightarrow {A_2} = 0.3 \times 1.02$
$\Rightarrow {A_2} = 0.306{m^2}$
Hence when a metal sheet of area $0.3{m^2}$ is heated from 293 K to $520^\circ C$, the area of metal sheet will be: ${A_2} = 0.306{m^2}$

Therefore, option (A) is the correct answer.

Note: When the temperature is increased the length, breadth, area and volume of a metal, is automatically increased. This process is called thermal expansion. Due to the thermal expansion of a ferromagnetic metal, it reduces its magnetization, and leads to its complete loss of its magnetism.