Dec 10, 2009 ... Part of Wangsness 11-3. A point charge q is located in the ... that your solution is
the same when you add 2π to ϕ. For simplicity, evaluate your.
PHYS333 Assignment # 11 Due Thursday, December 10, 2009 Reminders: Show your work! Include references on your submitted version. Write legibly! 1. Part of Wangsness 11-3 A point charge q is located in the xy plane near two grounded conducting planes as shown in Figure 11-14. The z axis lies along the intersection of the planes. Find and justify the image charges that, together with q, will give the potential at all points in the vacuum region x ≥ 0, y ≥ 0, −∞ < z < ∞. (Hint: recall multiple image formation in plane mirrors from geometrical optics.) Find φ(x, y, z) in the vacuum region. 2. Part of Wangsness 11-23 A spherical cavity of radius a is within a large grounded conductor. A charge q is placed within the cavity at a distance b from the center. Find φ at all points within the cavity (using the method of images) in spherical coordinates with origin at the center of the cavity and the z axis passing through the location of q. 3. A semi-infinite rectangular metal pipe with sides a and b is grounded. A separate piece of metal, insulated from the sides, is placed at the x = 0 end of the pipe and is maintained at a specified potential V0 (y, z), as indicated in the figure. Show that the potential everywhere inside the pipe is given by √ ∞ X ∞ X mπz nπy −x (nπ/a)2 +(mπ/b)2 Bm,n e φ(x, y, z) = sin sin a b m=1 n=1 where Bm,n
nπy mπz 4 Z bZ a V0 (y, z) sin sin dydz. = ab 0 0 a b
You may find the following integral useful: Z
0
c
!
c nπx n′ πx sin sin dx = δn,n′ c c 2
where δn,n′ is the Kroenecker delta with δn,n′ = 0 if n 6= n′ and δnn = 1.
4. Wangsness 11-24 The book’s hint is a little cryptic. While you’re not given any explicit boundary condition on φ, you know that the value at a particular radius ρ must be the same whether you define ϕ between 0 and 2π or from 2π to 4π, for example. Thus, you must enforce that your solution is the same when you add 2π to ϕ. For simplicity, evaluate your general solution at ϕ = 0 and ϕ = 2π and ensure they are always the same. 5. Part of Wangsness 11-25 Use your result from the previous problem to find the potential of an infinitely long cylindrical conductor with circular cross-section of radius a and axis along the z axis. Suppose it is placed in a previously uniform electric field E = E0 x ˆ (perpendicular to the cylinder!). Find the potential only outside the cylinder. (Don’t do the other parts of the book problem.)
