Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if:u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated. **Problem Statement:**Find the steady-state temperature $ u(r, \theta) $ in a semicircular plate of radius $ r = 2 $.**Boundary Conditions:**\[ u(2, \theta) = \begin{cases} 1 & \text{for } 0 < \theta < \pi/2 \\ 0 & \text{for } \pi/2 < \theta < \pi \end{cases}\]Additionally, the edges $ \theta = 0 $ and $ \theta = \pi $ are insulated. ---**Explanation:**The problem involves determining the temperature distribution, $ u(r, \theta) $, over a semicircular plate. The boundary conditions specify that along the curved edge of the semicircular plate (at $ r = 2 $), the temperature is set to 1 in the first quadrant ($ 0 < \theta < \pi/2 $) and 0 in the second quadrant ($ \pi/2 < \theta < \pi $). The straight edges of the semicircle, corresponding to $ \theta = 0 $ and $ \theta = \pi $, are insulated, meaning no heat flows across these boundaries.

Consider the given temperature ur, θ in a semi-circular plate of radius 2, and And the edges θ=0…

Answered: Find the steady-state temperature…

Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if: u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the steady-state temperature u(r,θ) in a semicirular plate of radius r=2 if:

u(2,θ) = {1, 0<θ<π/2 and 0, π/2<θ<π} and the edges θ = 0 and 0 = π are insuated.

**Problem Statement:**

Find the steady-state temperature $ u(r, \theta) $ in a semicircular plate of radius $ r = 2 $.

**Boundary Conditions:**

\[
u(2, \theta) =
\begin{cases}
1 & \text{for } 0 < \theta < \pi/2 \\
0 & \text{for } \pi/2 < \theta < \pi
\end{cases}
\]

Additionally, the edges $ \theta = 0 $ and $ \theta = \pi $ are insulated.

---

**Explanation:**

The problem involves determining the temperature distribution, $ u(r, \theta) $, over a semicircular plate. The boundary conditions specify that along the curved edge of the semicircular plate (at $ r = 2 $), the temperature is set to 1 in the first quadrant ($ 0 < \theta < \pi/2 $) and 0 in the second quadrant ($ \pi/2 < \theta < \pi $). The straight edges of the semicircle, corresponding to $ \theta = 0 $ and $ \theta = \pi $, are insulated, meaning no heat flows across these boundaries.

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