We perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section , as given by the Laplace equation .
This expansion , performed at the centre of the section , consists in an infinite series in the powers of the minor-to-major radius ratio e of the shell .
It is appropriate for a solid , homogeneous torus , as well as for inhomogeneous bodies ( the case of a core stratification is considered ) .
We show that the leading term is identical to the potential of a loop having the same main radius and the same mass — this “ similarity ” is shown to hold in the { \cal O } ( e ^ { 2 } ) order .
The series converges very well , especially close to the surface of the toroid where the average relative precision is \sim 10 ^ { -3 } for e = 0.1 at order zero , and as low as a few 10 ^ { -6 } at second order .
The Laplace equation is satisfied exactly in every order , so no extra density is induced by truncation .
The gravitational acceleration , important in dynamical studies , is reproduced with the same accuracy .
The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas .