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Question: When the temperature of a copper is raised by \({80^ \circ }C\), its diameter increases by \(0.2\% \...

When the temperature of a copper is raised by 80∘C{80^ \circ }C, its diameter increases by 0.2%0.2\% ,
A.Percentage rise in the area of a face is 0.4%0.4\%
B.Percentage rise in the thickness is 0.4%0.4\%
C.Percentage rise in the volume is 0.6%0.6\%
D.Co efficient of linear expansion of copper is 0.25×10−4/∘C0.25 \times {10^{ - 4}}{/^ \circ }C

Explanation

Solution

Recall the concept of linear expansion. Linear expansion means change in any one dimension or length to change in volume. It is defined as the fractional change in the length per degree of the change in temperature. It is caused due to change in temperature.

Complete answer:
Step I:
It is given that the diameter of the copper is increased. So the area will also be increased. The area expansion is written as
⇒ΔAA=2αΔT\Rightarrow \dfrac{{\Delta A}}{A} = 2\alpha \Delta T
Where α\alpha is the linear coefficient of thermal expansion.
And ΔT\Delta Tis the change in temperature.
⇒ΔAA=2×LinearExpansion\Rightarrow \dfrac{{\Delta A}}{A} = 2 \times LinearExpansion
⇒ΔAA=2×0.2\Rightarrow \dfrac{{\Delta A}}{A} = 2 \times 0.2
⇒ΔAA=0.4%\Rightarrow \dfrac{{\Delta A}}{A} = 0.4\%
Step II:
Due to change in area, its volume will also change. The change in volume or volume expansion is given by
⇒ΔVV=3×LinearExpansion\Rightarrow \dfrac{{\Delta V}}{V} = 3 \times LinearExpansion
⇒ΔVV=3×0.2\Rightarrow \dfrac{{\Delta V}}{V} = 3 \times 0.2
⇒ΔVV=0.6%\Rightarrow \dfrac{{\Delta V}}{V} = 0.6\%
Step III:
The change in diameter will be written as
⇒ΔDD=αΔT\Rightarrow \dfrac{{\Delta D}}{D} = \alpha \Delta T
⇒ΔDD=0.2%\Rightarrow \dfrac{{\Delta D}}{D} = 0.2\%
Given ΔT=80∘C\Delta T = {80^ \circ }C
Therefore it can be written that
⇒0.2%=α.80\Rightarrow 0.2\% = \alpha .80
⇒α=0.00280\Rightarrow \alpha = \dfrac{{0.002}}{{80}}
Or ⇒α=2080×104 \Rightarrow \alpha = \dfrac{{20}}{{80}} \times {10^4}
⇒α=0.25×10−4\Rightarrow \alpha = 0.25 \times {10^{ - 4}}
Step IV:
Therefore, it is clear that the rise in the area of the face of the copper wire is 0.4%0.4\%
Percentage rise in the volume of copper is 0.6%0.6\%
Coefficient of linear expansion of copper is 0.25×10−4/∘C−10.25 \times {10^{ - 4}}{/^ \circ }{C^{ - 1}}
Step V:

Hence Option A, C and D is the right answer.

Note:
It is to be noted that the change in length of any metal or object will be directly proportional to the change in temperature. This means if an object is heated or cooled, then the change in length is given by original length and change in temperature. The objects expand on heating, because the molecules gain energy and start moving faster. Also this causes change in volume and they occupy more space.