{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Integration with variables Notes_Part_15

# Integration with variables Notes_Part_15 - Figure 7.25 A...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Figure 7.25. A Non–Coaxial Cable. Example 7.39. A non-coaxial cable . The goal of this example is to determine the electrostatic potential inside a non-coaxial cylindrical cable, as illustrated in Figure 7.25, with prescribed constant potential values on the two bounding cylinders. Assume for definiteness that the larger cylinder has radius 1, and is centered at the origin, while the smaller cylinder has radius 2 5 , and is centered at z = 2 5 . The resulting electrostatic potential will be independent of the longitudinal coordinate, and so can be viewed as a planar potential in the annular domain contained between two circles representing the cross-sections of our cylinders. The desired potential must satisfy the Dirichlet boundary value problem Δ u = 0 when | z | < 1 and vextendsingle vextendsingle z − 2 5 vextendsingle vextendsingle > 2 5 , u = a, when | z | = 1 , and u = b when vextendsingle vextendsingle z − 2 5 vextendsingle vextendsingle = 2 5 . According to Example 7.36, the linear fractional transformation ζ = 2 z − 1 z − 2 (7 . 92) maps this non-concentric annular domain to the annulus A 1 / 2 , 1 = braceleftbig 1 2 < | ζ | < 1 bracerightbig , which is the cross-section of a coaxial cable. The corresponding transformed potential U ( ξ, η ) has the constant Dirichlet boundary conditions U = a, when | ζ | = 1 2 , and U = b when | ζ | = 1 . (7 . 93) Clearly the coaxial potential U must be a radially symmetric solution to the Laplace equation, and hence, according to (6.103), of the formequation, and hence, according to (6....
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

Integration with variables Notes_Part_15 - Figure 7.25 A...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online