Consider an infinite square pipe of size 4 mx4 m with the bottom three faces held...

 

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Consider an infinite square pipe of size 4 mx4 m with the bottom three faces held at V = 0 and the top face held at V = 1 V. Discretize this as a grid (x, y = 0, 1, 2, 3, 4), such that the potentials and charge densities at each grid point are Hello square Pipe 0 1 I 0 ? ? ? ? ? 0 V13 V23 V33 0 ? 0 0 0 ? V(x, y)=Vxy=0 V12 V22 V32 0 p(x, y)=? 0 0 0 ? 0 V1 V2 V31 0 ? 0 0 0 ? 00 0 0 0 ? ? ? ? ? (a) Write down 9 equations, one for each unknown potential. For example, V13 satisfies 4V13 - V23-1-V12=0. (b) Write these simultaneous equations as a matrix equation of the form below. The row corresponding to 13 has been filled in for you as an example. ㅁ 口口 ☐ ㅁ DOD ㅁ ㅁ ㅁ ☐ ם ם ם ם ם ם י ם ם V11 V12 0-1 4 0 0 -1 0 0 0 V13 V21 V22 D V23 ㅁ V31 ☐ 品 OOOOOOOO 1 ☐ ㅁㅁ V32 V33 (c) Solve the 9×9 linear system. You could do this by hand using Gaussian elimination, but you are advised to use LinearSolve [] in Mathematica or SOL-RHS/MATRIX in MATLAB. (d) Compare your approximate numerical results at each grid point with the analytical solution obtained from separation of variables in Cartesians, 4 V(x, y) = T Σ-1,3.5.... плх sin “ sinh "ay n sinh n 4 (0 <x<4 and 0 < y <4). Consider an infinite square pipe of size 4 mx4 m with the bottom three faces held at V = 0 and the top face held at V = 1 V. Discretize this as a grid (x, y = 0, 1, 2, 3, 4), such that the potentials and charge densities at each grid point are Hello square Pipe 0 1 I 0 ? ? ? ? ? 0 V13 V23 V33 0 ? 0 0 0 ? V(x, y)=Vxy=0 V12 V22 V32 0 p(x, y)=? 0 0 0 ? 0 V1 V2 V31 0 ? 0 0 0 ? 00 0 0 0 ? ? ? ? ? (a) Write down 9 equations, one for each unknown potential. For example, V13 satisfies 4V13 - V23-1-V12=0. (b) Write these simultaneous equations as a matrix equation of the form below. The row corresponding to 13 has been filled in for you as an example. ㅁ 口口 ☐ ㅁ DOD ㅁ ㅁ ㅁ ☐ ם ם ם ם ם ם י ם ם V11 V12 0-1 4 0 0 -1 0 0 0 V13 V21 V22 D V23 ㅁ V31 ☐ 品 OOOOOOOO 1 ☐ ㅁㅁ V32 V33 (c) Solve the 9×9 linear system. You could do this by hand using Gaussian elimination, but you are advised to use LinearSolve [] in Mathematica or SOL-RHS/MATRIX in MATLAB. (d) Compare your approximate numerical results at each grid point with the analytical solution obtained from separation of variables in Cartesians, 4 V(x, y) = T Σ-1,3.5.... плх sin “ sinh "ay n sinh n 4 (0 <x<4 and 0 < y <4). Consider an infinite square pipe of size 4 mx4 m with the bottom three faces held at V = 0 and the top face held at V = 1 V. Discretize this as a grid (x, y = 0, 1, 2, 3, 4), such that the potentials and charge densities at each grid point are Hello square Pipe 0 1 I 0 ? ? ? ? ? 0 V13 V23 V33 0 ? 0 0 0 ? V(x, y)=Vxy=0 V12 V22 V32 0 p(x, y)=? 0 0 0 ? 0 V1 V2 V31 0 ? 0 0 0 ? 00 0 0 0 ? ? ? ? ? (a) Write down 9 equations, one for each unknown potential. For example, V13 satisfies 4V13 - V23-1-V12=0. (b) Write these simultaneous equations as a matrix equation of the form below. The row corresponding to 13 has been filled in for you as an example. ㅁ 口口 ☐ ㅁ DOD ㅁ ㅁ ㅁ ☐ ם ם ם ם ם ם י ם ם V11 V12 0-1 4 0 0 -1 0 0 0 V13 V21 V22 D V23 ㅁ V31 ☐ 品 OOOOOOOO 1 ☐ ㅁㅁ V32 V33 (c) Solve the 9×9 linear system. You could do this by hand using Gaussian elimination, but you are advised to use LinearSolve [] in Mathematica or SOL-RHS/MATRIX in MATLAB. (d) Compare your approximate numerical results at each grid point with the analytical solution obtained from separation of variables in Cartesians, 4 V(x, y) = T Σ-1,3.5.... плх sin “ sinh "ay n sinh n 4 (0 <x<4 and 0 < y <4).

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Write down the equation for V11 using the given example and the fact that the potential at a point i... View the full answer