  We combine and further develop ideas and techniques of Allen \& Ottewill,
Phys. Rev.D, {\bf 42}, 2669 (1990) and Kay \& Studer Commun. Math. Phys., {\bf
139}, 103 (1991) for calculating the long range effects of cosmic string cores
on classical and quantum field quantities far from an (infinitely long,
straight) cosmic string. We find analytical approximations for (a) the
gravity-induced ground state renormalized expectation values of $\hat\varphi^2$
and $\hat T_\mu{}^\nu$ for a non-minimally coupled quantum scalar field far
from a cosmic string (b) the classical electrostatic self force on a test
charge far from a superconducting cosmic string. Surprisingly -- even at
cosmologically large distances -- all these quantities would be very badly
approximated by idealizing the string as having zero thickness and imposing
regular boundary conditions; instead they are well approximated by suitably
fitted strengths of logarithmic divergence at the string core. Our formula for
${\langle {\hat \varphi}^2 \rangle}$ reproduces (with much less effort and much
more generality) the earlier numerical results of Allen \& Ottewill. Both
${\langle {\hat \varphi}^2 \rangle}$ and ${\langle {\hat T}_{\mu}{}^{\nu}
\rangle}$ turn out to be ``weak field topological invariants'' depending on the
details of the string core only through the minimal coupling parameter
``$\xi$'' (and the deficit angle). Our formula for the self-force (leaving
aside relatively tiny gravitational corrections) turns out to be attractive: We
obtain, for the self-potential of a test charge $Q$ a distance $r$ from a (GUT
scale) superconducting string, the formula $- Q^2/(16\epsilon_0r\ln(qr))$ where
$q$ is an (in principle, computable) constant of the order of the inverse
string radius.
