Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities ρ(x), where x is the position vector.
(a) In spherical coordinates, a charge Q uniformly distributed over a spherical shell of radius R.
(b) In cylindrical coordinates a charge λ per unit length uniformly distributed over a cylindrical surface of radius b.
(c) In cylindrical coordinates, a charge Q spread uniformly over a flat circular disc of negligible thickness and radius R.
Solution:
(a) If the center of the spherical coordinate system (r, θ, φ) is based on the center of the spherical shell, charge is distributed at r=R uniformly, and this distribution of charge is symmetric about θ and φ coordinates ( which are polar and azimuthal angles). So distribution of charge varies only with respect to r, and it can be represented as,
ρ=C δ(r-R)
where ρ is the density of charge, and C is a constant that we want to calculate it, and δ(r-R) is Dirac delta function defined as,
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