Consider an infinite square pipe of size 4 mx4 m with the bottom three faces held...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
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).
Expert Answer:
Answer rating: 100% (QA)
Write down the equation for V11 using the given example and the fact that the potential at a point i... View the full answer
Related Book For
Numerical Methods With Chemical Engineering Applications
ISBN: 9781107135116
1st Edition
Authors: Kevin D. Dorfman, Prodromos Daoutidis
Posted Date:
Students also viewed these law questions
-
An m n matrix has full row rank if its row rank is m, and it has full column rank if its column rank is n. (a) Show that a matrix can have both full row rank and full column rank only if it is...
-
Write down the elementary matrix corresponding to the following row operations on 4 4 matrices: (a) Add the third row to the fourth row. (b) Subtract the fourth row from the third row. (c) Add 3...
-
Write down a real matrix that has (a) Eigenvalues -1, 3 and corresponding eigenvectors (b) Eigenvalues 0, 2, -2 and associated eigenvecors (c) An eigenvalue of 3 and corresponding eigenvectors (d) An...
-
A line charge density pL of length L lies parallel to an infinite sheet of surface charge density ps. How much work is required to rotate the line charge so that it is vertical?
-
The Old School is trying to decide whether they should purchase or lease a new high-speed photocopier machine. It will cost the school $ 14,000 to purchase the copier and $ 750 each year to maintain...
-
What are the most important problems you see with being a digital nomad?
-
Epic Systems is a Wisconsin health care software company. In 2014, Epic introduced a company policy that required employees to use individual arbitration in any disputes. Jacob Lewis, an Epic...
-
On December 21, 2012, Zurich Company provided you with the following information regarding its trading investments. During 2013, Carolina Company shares were sold for $9,500. The fair value of the...
-
a. Write a story problem for the expression 6 x b. Find the value of the expression. Show or explain your work. da f
-
Reba Dixon is a fifth-grade school teacher who earned a salary of $38,000 in 2020. She is 45 years old and has been divorced for four years. She receives $1,200 of alimony payments each month from...
-
E14.2 (LO 1, 2) Video Finance (Information Related to Various Bond Issues) Anaconda Inc. has issued three types of debt on January 1, 2023, the start of the company's fiscal year: 1. $10 million,...
-
Prepare the general journals to close the accounts of J Smith (shown in figure 9.26) for the year ended 30 June 2022. Extract Account Balances of J Smith as at 30 June 2022 Sales Cost of Sales...
-
Wakefield Heights sold $6,000,000 of general obligation serial bonds at a 1.5% discount to finance the construction of a new recreation center. Bond issuance costs were 2% of the face amount of the...
-
A pressurized propane tank is fitted with a valve on top that allows saturated vapor to escape when the valve is opened. When the valve is first opened, the mass flow rate of the propane leaving the...
-
At the financial years end, prepare the general journal to close out the accounts to the profit and loss account in the general ledger and post the journal to the T accounts shown in figure 9.20....
-
From the account balances of R Jones in figure 9.25, prepare the necessary closing general journals. Extract Account Balances of R Jones as at 31 December 2022 Inventory [1-1-2022] Assets - other...
-
XPEL Inc, has a target capital structure of 30% of preferred stock and the remaining equally weighed with common stock and debt. The relevant tax fate is 31%. The bonds were issued with 5% coupon...
-
Draw the major product for each of the following reactions: (a) (b) (c) 1) 9-BBN 2) H2O2, NaOH 1) Disiamylborane 2) H20, NaOH
-
Consider the solution to the diffusion equation on a 4 4 rectangular grid with equal spacing in the x and y directions. The boundary conditions for the problem are Using the global index k and the...
-
Solve the same problem as Example 3.13 with the equations in the opposite order. Data from Example 3.13 Solve the system of Eqs. (3.7.6) and (3.7.7) using Newtons method with an initial guess x (0) 1...
-
Use RK4 to find the solution to subject to the initial conditions y(0) = 1, y(0) = 1/2, y(0) = 1/3, y= 1/4, and y = 1/5. Your program should make a plot of the fourth derivative of y versus the...
-
Calculate (in terms of a single ket state and no operators) exp{ila a+ B(a+a)]}|a). (8.108)
-
Calculate \(\left\langle x^{2} ightangle_{n}\) and \(\left\langle x^{3} ightangle_{n}\) in the state \(|nangle\) of the harmonic oscillator.
-
Use the Sommerfeld method for a particle in one dimension with a quartic potential, instead of a quadratic one, \(V(x)=\lambda x^{4}\). What is the resulting reduced equation, and can you describe...
Study smarter with the SolutionInn App