a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σ, that is lying in the xy plane centered on the orign. Using the Coulomb formula for the potential, (r)-da σ(r') |r-r' do the integral to find the exact value of the potential o for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the following parts!). b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential o(r) in terms of a Legendre polynomial series using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order ¤=5, for |r|>R and |r|
a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σ, that is lying in the xy plane centered on the orign. Using the Coulomb formula for the potential, (r)-da σ(r') |r-r' do the integral to find the exact value of the potential o for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the following parts!). b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential o(r) in terms of a Legendre polynomial series using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order ¤=5, for |r|>R and |r|
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Transcribed Image Text:a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σ, that is lying in the xy plane centered on the orign. Using the Coulomb formula for
the potential,
(r)-da
σ(r')
|r-r'
do the integral to find the exact value of the potential o for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the
following parts!).
b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential o(r) in terms of a Legendre polynomial series
using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree
with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order ¤=5, for |r|>R and |r|<R, both above and below the disk.
c) Using your results of part (b), compute the electric field E just above and just below the disk, and explicitly show that the discontinuity in E is given by the surface charge density
σo, as we know from general principles that it must be.
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