Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). Write down the appropriate Green function G(x, x').

Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). Write down the appropriate Green function G(x, x').

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). 2.7 (a) Write down the appropriate Green function G(x, x'). (b) If the potential on the plane z = 0 is specified to be = V inside a circle of radius a centered at the origin, and = 0 outside that circle, find an integral expression for the potential at the point P specified in terms of cylindrical coordinates (p, 4, z). (c) Show that, along the axis of the circle (p = 0), the potential is given by O = VI 1 - Va? + (d) Show that at large distances (p? + z2 > a?) the potential can be expanded in a power series in (p? + z?)-1, and that the leading terms are Va? 3a? 5(3p?a? + a*) + +.. 2 (p? + z33/2 4(p2 + z?) 8(p2 + z?)? Verify that the results of parts c and d are con sistent with each other in their common range of validity.
Transcribed Image Text:Consider a potential problem in the half-space defined by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at infinity). 2.7 (a) Write down the appropriate Green function G(x, x'). (b) If the potential on the plane z = 0 is specified to be = V inside a circle of radius a centered at the origin, and = 0 outside that circle, find an integral expression for the potential at the point P specified in terms of cylindrical coordinates (p, 4, z). (c) Show that, along the axis of the circle (p = 0), the potential is given by O = VI 1 - Va? + (d) Show that at large distances (p? + z2 > a?) the potential can be expanded in a power series in (p? + z?)-1, and that the leading terms are Va? 3a? 5(3p?a? + a*) + +.. 2 (p? + z33/2 4(p2 + z?) 8(p2 + z?)? Verify that the results of parts c and d are con sistent with each other in their common range of validity.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Triple Integral
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS