Question

Find the steady-state temperature distribution in a rectangular plate covering the area \(0

Short answer

The steady-state temperature distribution is given by T(x,y) = xy.

Step-by-step solution

Set up the boundary conditions

The plate is defined by the region where 0 < x < 10 0 < y < 20 . We have the following boundary conditions: Along the sides where x=0 and x=10, the temperatures are T=y and T=0 respectively. Along the sides where y=0 and y=20, the temperatures are T=x and T=0 respectively.

Write the general solution for the Laplace equation

Using separation of variables, the general solution in two dimensions for the steady-state heat equation ∇²T = 0 can be written as T(x,y) = X(x)Y(y).

Apply boundary conditions to solve for X(x) and Y(y)

Applying the boundary condition at x=0 (T=y) and x=10 (T=0) tells us: X(0)Y(y)=y X(10)Y(y)=0 This suggests a linear variation for X(x), hence X(x)=A(x)=x.

Solve for the solution

Substitute X(x)=x: T(x,y) = xY(y) Using the boundary condition T=0 along y=20 we get: xY(20)=0, thus Y(20)=0 This implies a linear variation for Y(y), hence Y(y)=y, and T(x,y) = xy.

Key concepts

Laplace Equation

The Laplace Equation is a second-order partial differential equation named after the mathematician Pierre-Simon Laplace. It is written as \( abla^2 T = 0 \), where \( abla^2 \) is the Laplacian operator. This equation is fundamental in the study of steady-state temperature distributions because it describes scenarios where there are no internal heat sources or sinks.

In the context of the exercise, solving the Laplace equation helps us determine the temperature distribution on the plate, assuming heat has evenly diffused across the surface. The two-dimensional version of this equation is \( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0 \). This simplifies temperature distribution analysis by breaking it down into more manageable calculations using boundary conditions and specific methods like separation of variables.

Boundary Conditions

Boundary conditions are constraints necessary to solve differential equations like the Laplace Equation. They represent the physical conditions or constraints at the edges of the region we are studying. For example, in the exercise, the temperatures along the edges of the plate are specified.

The boundary conditions for the given rectangular plate are:
- Along \( x=0 \): \( T=y \)
- Along \( x=10 \): \( T=0 \)
- Along \( y=0 \): \( T=x \)
- Along \( y=20 \): \( T=0 \)
These conditions are crucial for determining the functions that satisfy the Laplace Equation within the region. By applying these conditions, we are able to constrain possible solutions and find the exact temperature distribution.

Separation Of Variables

Separation of Variables is a powerful method for solving partial differential equations like the Laplace Equation. The idea is to assume that the function we are looking for can be written as a product of functions, each depending on a single variable.

In our exercise, we assume that \( T(x,y) = X(x)Y(y) \), where \( X \) is a function of \( x \) and \( Y \) a function of \( y \). This simplifies the two-dimensional problem into two one-dimensional problems. Applying the boundary conditions progressively yields:
- \( X(x) = x \)
- \( Y(y) = y \)

Thus, the solution to the two-dimensional Laplace Equation is the product \( T(x,y) = xy \).

Steady-State Heat Equation

The steady-state heat equation is another term for the Laplace Equation when used to describe heat distribution without time variation. This means the system has reached a point where temperature does not change over time, only in space.

For the plate, we assume it's in a steady state (no heat flow over time), which leads to \( abla^2 T = 0 \). This simplifies analysis and allows us to focus on spatial temperature variations only. The steady-state solution provides a snapshot of how heat distributes across the plate under the given boundary conditions. For the given exercise, solving the steady-state heat equation with the boundary conditions reveals the final temperature distribution \( T(x,y) = xy \).