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 $\hatφ^2$ and $\hat T_μ^ν$ 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 ${\langle {\hat φ}^2 \rangle}$ reproduces (with much less effort and much more generality) the earlier numerical results of Allen \& Ottewill. Both ${\langle {\hat φ}^2 \rangle}$ and ${\langle {\hat T}_μ{}^ν \rangle}$ turn out to be ``weak field topological invariants'' depending on the details of the string core only through the minimal coupling parameter ``$ξ$'' (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 $- Q^2/(16ε_0r\ln(qr))$ where $q$ is an (in principle, computable) constant of the order of the inverse string radius.

25 pages, RevTeX with epsfig, 3 Postscript figures