Chapter 13: Q24MP (page 664)

Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc. $T = A, T = B$ , on the four sides, and the rectangle covers the area $0 < x < a, 0 < y < b$ .

Short Answer

Expert verified

The general solution for the steady-state temperatureis $T = \sum_{n = 1}^{\infty} b_{n} \sin (\frac{n π x}{10})$ .

Step by step solution

Given information:

To find the general solution for the steady state temperature.

And the figure 2.2 is shown below:

Definition of Steady state temperature:

When a conductor reaches a point where no more heat can be absorbed by the rod, it is said to be at steady-state.

Step 3:Use Boundary conditions

Consider the boundary condition $T = (\begin{matrix} A 0 < x < a \\ B 0 < y < b \end{matrix})$ .

The given figure is:

Write the basis solution of the boundary condition.

$T = (\begin{matrix} e^{k y} \sin k x \\ e^{- k y} \sin k x \\ e^{k y} \cos k x \\ e^{- k y} \cos k x \end{matrix})$

The above basis solution satisfies the given boundary conditions.

Find the temperature for $0 < x < a$ .

$T = A e^{- k a} \sin (k a)$

Find the temperature for $0 < y < b$ .

$T = B e^{- k b} \sin (k b)$

The solution is given by,

$T = \sum_{n = 1}^{\infty} b_{n} \sin (\frac{n π x}{a})$

If, $T = (\begin{matrix} A 0 < x < a \\ B 0 < y < b \end{matrix})$

Then

$T = \sum_{n = 1}^{\infty} e^{\frac{n π y}{a}} b_{n} \sin (\frac{n π x}{a}) = B$ .

Hence the general solution for the steady-state temperatureis $T = \sum_{n = 1}^{\infty} b_{n} \sin (\frac{n π x}{10})$ .

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Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc. $T = A, T = B$ , on the four sides, and the rectangle covers the area $0 < x < a, 0 < y < b$ .

Short Answer

Step by step solution

Given information:

Definition of Steady state temperature:

Step 3:Use Boundary conditions

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Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc.T=A, T=B, on the four sides, and the rectangle covers the area 0<x<a, 0<y<b.

Short Answer

Step by step solution

Given information:

Definition of Steady state temperature:

Step 3:Use Boundary conditions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Engineering Physics

Radiation

Famous Physicists

Astrophysics

Oscillations

Study anywhere. Anytime. Across all devices.

Find the general solution for the steady-state temperature in Figure 2.2 if the boundary temperatures are the constants, etc. $T = A, T = B$ , on the four sides, and the rectangle covers the area $0 < x < a, 0 < y < b$ .