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Unformatted text preview: HW 6 Phys5406 S03 due 3/27/03
1) The object is to ﬁnd the Green’s function inside a cylindrical volume using, in part, the
solution to the cylindrical problem of JDJ. This is a cylinder of radius a and length L. All
surfaces are grounded except that at z=L and it is kept at potential V(ρ, φ). The solution
is in JDJ and was worked out in class.
a) Write the solution for the potential in terms of an integral involving the derivative of GD .
This should indicate what terms are required in GD aside from constants you will determine
b) Recall the result for sinh γz /sinh γL expanded in a sine series and that for the sine series
expansion of δ (z − z ).
c) Show that δ (φ − φ ) = AΣm eim(φ−φ ) and ﬁnd A.
d) Show that δ ( ρ− ρ )
ρ e) Noting that 2 = Σn Bn Jm ( χmn ρ )Jm ( χmn ρ ) and ﬁnd Bn .
a GD = −4πδ (r − r ), put these pieces together and obtain GD . 2) Solve the problem when the surfaces at z=0 and z=L are at constant potential V and
the round surface of the cylinder is grounded. You can use either the Green’s function or
directly from LaPlace’s equation. For practise, the adventurous can show that both give the
3) A spherical volume of radius a has a volume charge density ρ = Qx/a4 . Find the potential
inside and outside of the sphere. Hint use spherical harmonics and the spherical addition
theorem. 1 ...
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