To understand how the area of a rectangular plate changes with temperature, we need to delve into the concept of linear expansion. When a material is heated, its dimensions increase. For a rectangular plate with sides 'a' and 'b', the area 'A' is given by the product of these two sides: A = ab. When the temperature rises by a certain amount, denoted as ∆T, both sides will expand, leading to a change in area, which we want to express mathematically.

Understanding Linear Expansion

The linear expansion of a material is described by the formula:

∆L = αL₀∆T

Here, ∆L is the change in length, α is the coefficient of linear expansion, L₀ is the original length, and ∆T is the change in temperature. For our rectangular plate:

• For side 'a': ∆a = αa∆T
• For side 'b': ∆b = αb∆T

Calculating the New Area

After the temperature increase, the new lengths of the sides will be:

• New length of side 'a': a + ∆a = a + αa∆T = a(1 + α∆T)
• New length of side 'b': b + ∆b = b + αb∆T = b(1 + α∆T)

Thus, the new area A' can be expressed as:

A' = (a + ∆a)(b + ∆b)

Substituting the expressions for ∆a and ∆b, we get:

A' = a(1 + α∆T) b(1 + α∆T)

Expanding this, we find:

A' = ab(1 + α∆T)(1 + α∆T) = ab(1 + 2α∆T + (α∆T)²)

Finding the Change in Area

The change in area, ∆A, is given by:

∆A = A' - A = ab(1 + 2α∆T + (α∆T)²) - ab

By simplifying this expression, we have:

∆A = ab(2α∆T + (α∆T)²)

Neglecting the Small Quantity

Since we are instructed to neglect the small quantity (α∆T)², we can simplify our expression further:

∆A ≈ 2αA∆T

This shows that the change in area is approximately twice the product of the coefficient of linear expansion, the original area, and the change in temperature.

Final Verification

Thus, we have successfully demonstrated that if we neglect the small quantity ∆a∆b, the change in area of the rectangular plate due to a temperature rise ∆T can be expressed as:

∆A = 2αA∆T

This relationship highlights the significant effect of temperature on the dimensions of materials, which is crucial in various engineering and physical applications.