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A physics homework assignment from the university of california, berkeley, for a fall 2005 class (physics 505). The assignment includes problems related to gauss' theorem, electrostatic potential, and green's functions. Students are required to use the given textbook problems (chapters 1, 1.4, 1.5, 1.10, 1.14) to find electric fields, charge distributions, and apply the mean value theorem and green's theorem.
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Physics 505 Fall 2005 Homework Assignment #1 — Due Thursday, September 15
Textbook problems: Ch. 1: 1.4, 1.5, 1.10, 1.
1.4 Each of three charged spheres of radius a, one conducting, one having a uniform charge density within its volume, and one having a spherically symmetric charge density that varies radially as rn^ (n > −3), has a total charge Q. Use Gauss’ theorem to obtain the electric fields both inside and outside each sphere. Sketch the behavior of the fields as a function of radius for the first two spheres, and for the third with n = −2, +2.
1.5 The time-averagesd potential of a neutral hydrogen atom is given by
q 4 π 0
e−αr r
αr 2
where q is the magnitude of the electronic charge, and α−^1 = a 0 /2, a 0 being the Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically.
1.10 Prove the mean value theorem: For charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.
1.14 Consider the electrostatic Green functions of Section 1.10 for Dirichlet and Neumann boundary conditions on the surface S bounding the volume V. Apply Green’s theorem (1.35) with integration variable ~y and φ = G(~x, ~y ), ψ = G(~x ′, ~y), with ∇^2 y G(~z, ~y ) = − 4 πδ(~y − ~z ). Find an expression for the difference [G(~x, ~x ′) − G(~x ′, ~x)] in terms of an integral over the boundary surface S. a) For Dirichlet boundary conditions on the potential and the associated boundary condition on the Green function, show that GD (~x, ~x ′) must be symmetric in ~x and ~x ′. b) For Neumann boundary conditions, use the boundary condition (1.45) for GN (~x, ~x ′) to show that GN (~x, ~x ′) is not symmetric in general, but that GN (~x, ~x ′) − F (~x ) is symmetric in ~x and ~x ′, where
F (~x ) =
S
GN (~x, ~y ) day
c) Show that the addition of F (~x ) to the Green function does not affect the potential Φ(~x ). See problem 3.26 for an example of the Neumann Green function.