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Question 2. Spherical boundaries call for spherical coordinates, and in this problem, we take on the task of solving Laplace's equation in these coordinates. The Laplacian operator expressed in terms of (r, 0, q) 0 = V²0 = 2 (200) + ² sino 30 ( 1 1 ə r² dr Ꮎ ᎧᎾ b 0 = A a V = Vo V = 0 a sin 0 аф де Y + 1 Φ r2 sin² 0 дф2 Figure 1: Laplace equation bounded by a rectangular conducting box of width a and height b. All sides of the box are maintained at = 0 except for the top side, which is maintained at = $0. (a) We shall separate the scalar field Þ= Þ(r, 0, q) into two parts; a radial function R and an angular function Y known as spherical harmonics, so that Þ(r, 0, p) = R(r)Y(0, p). Use separation of variables to express equation into two equations, using a separation constant 2.

Question 2. Spherical boundaries call for spherical coordinates, and in this problem, we take on the task of solving Laplace's equation in these coordinates. The Laplacian operator expressed in terms of (r, 0, q) 0 = V²0 = 2 (200) + ² sino 30 ( 1 1 ə r² dr Ꮎ ᎧᎾ b 0 = A a V = Vo V = 0 a sin 0 аф де Y + 1 Φ r2 sin² 0 дф2 Figure 1: Laplace equation bounded by a rectangular conducting box of width a and height b. All sides of the box are maintained at = 0 except for the top side, which is maintained at = $0. (a) We shall separate the scalar field Þ= Þ(r, 0, q) into two parts; a radial function R and an angular function Y known as spherical harmonics, so that Þ(r, 0, p) = R(r)Y(0, p). Use separation of variables to express equation into two equations, using a separation constant 2.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 97E
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Question 2. Spherical boundaries call for spherical coordinates, and in this problem, we take on the task of solving Laplace's equation in these coordinates. The Laplacian operator expressed in terms of (r, 0, p) 0 = V²0 = 10 (200) + 7² sine 30 ( 1 ə r² dr b V = 0 a V = Vo V = 0 a sin 0 дф де Y + 1 Φ r2 sin² 0 дф2 Figure 1: Laplace equation bounded by a rectangular conducting box of width a and height b. All sides of the box are maintained at Þ= 0 except for the top side, which is maintained at = 0. (a) We shall separate the scalar field Þ= (r, 0, q) into two parts; a radial function R and an angular function Y known as spherical harmonics, so that Þ(r, 0, p) = R(r)Y(0, p). Use separation of variables to express equation into two equations, using a separation constant 2.
Transcribed Image Text:Question 2. Spherical boundaries call for spherical coordinates, and in this problem, we take on the task of solving Laplace's equation in these coordinates. The Laplacian operator expressed in terms of (r, 0, p) 0 = V²0 = 10 (200) + 7² sine 30 ( 1 ə r² dr b V = 0 a V = Vo V = 0 a sin 0 дф де Y + 1 Φ r2 sin² 0 дф2 Figure 1: Laplace equation bounded by a rectangular conducting box of width a and height b. All sides of the box are maintained at Þ= 0 except for the top side, which is maintained at = 0. (a) We shall separate the scalar field Þ= (r, 0, q) into two parts; a radial function R and an angular function Y known as spherical harmonics, so that Þ(r, 0, p) = R(r)Y(0, p). Use separation of variables to express equation into two equations, using a separation constant 2.
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