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The air-core Barrel Toroid of the ATLAS Detector at the CERN Large Hadron Collider, is the largest superconducting magnet ever built. This muon-spectrometer magnet consists of 8 discrete coils, arranged symmetrically around the beam axis. The average field is 0.5 T, and the stored energy 1.1 GJ. The windings are 25 m long, 5 m wide and 0.4 m thick. Each coil cold mass consists of 2 double pancakes of aluminum-stabilized NbTi conductor securely fixed in an aluminum alloy casing. Before assembly, each coil is tested individually in the presence of a magnetic mirror to simulate the magnetic forces exerted on the tie rods. This mirror is composed of several iron plates made of magnetically non-linear material. Each coil is enclosed in a separate, stainless-steel cryostat with relative magnetic permeability slightly larger than 1. In this paper the magnetic field distribution and Lorenz forces between the magnetic mirror and the coil in centered and shifted positions are presented. The magnetic influence of the cryostat on the field distribution is also discussed.