A thin copper wire of length L increases in length by 1% when heated from temperature T1 to T2. What is the percentage change in area when a thin copper plate having dimensions 2L × L is heated from T1 to T2?
Question
A thin copper wire of length L increases in length by 1% when heated from temperature T1 to T2. What is the percentage change in area when a thin copper plate having dimensions 2L × L is heated from T1 to T2?
Solution
The percentage change in length due to thermal expansion is given by the formula:
ΔL/L = α * ΔT
where: ΔL is the change in length, L is the original length, α is the coefficient of linear expansion, ΔT is the change in temperature.
From the problem, we know that ΔL/L = 1%.
Now, let's consider the change in area of the copper plate. The area of the plate before heating is A1 = 2L * L = 2L^2. When the plate is heated, each dimension increases by 1%, so the new dimensions are 2L * 1.01 and L * 1.01. Therefore, the area of the plate after heating is A2 = 2L * 1.01 * L * 1.01 = 2.0402L^2.
The percentage change in area is given by the formula:
ΔA/A = (A2 - A1) / A1 * 100%
Substituting the values we found:
ΔA/A = (2.0402L^2 - 2L^2) / 2L^2 * 100% = 2.01%
So, the percentage change in area when the copper plate is heated from T1 to T2 is approximately 2.01%.
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