Single-Particle Electrodynamics - Assassination Science

One can very simply verifiy that the stated result (5.81) is correct for
a uniformly charged sphere, by taking its divergence and curl. Noting that
the expression (5.81) is continuous everywhere (in particular, there are no
abrupt jumps anywhere on the surface of the sphere, where the functional
form of the result changes), and making use of the identities of Section B.3,
one finds
⎧
⎪⎨
∇·E q (r) =
⎪⎩
3q
4πε 3 , r < ε,
0, r > ε,
(5.82)
which is of course recognised as the correct density of electric charge, when
one recalls that the volume of the sphere is 4πε 3 /3. The curl of (5.81) is, of
course, zero.
(The boundary condition implicitly assumed in obtaining (5.81) as the
integral of (5.82) is, of course, that the fields vanish at spatial infinity.)
5.5.2 The fields from an electric dipole
For a spherical body throughout which an electric dipole moment density is
uniformly distributed, we may obtain the static field most simply by noting
that each elementary constituent dipole is equivalent to two equal and
opposite electric charges placed infinitesimally close together, aligned in the
direction of d; and that all of the d three-vectors of the constituent dipoles
are coherently aligned, when the body as a whole is static.
This means
that we may simply superpose the fields of the constituents according to the
relation valid for each:
E d (r) = − 1 q (d ·∇)Eq (r).
(The minus sign appears because by separating the charges by an infinitesimal
amount we are actually shifting the origin of coördinates, not the position
where we measure the field.) By obtaining the gradient of the charge
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