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The diffusion of an external magnetic field through a toroidal conducting shell is studied under the assumption of a small aspect ratio. The external magnetic field can have an arbitrary field index and magnetic flux on the minor axis of the torus. The diffused field, field index, magnetic flux, and wall current are computed analytically and compared with the numerical results from the TRIDIF code. The analytical and numerical results are in good agreement. Measurements in the toroidal chamber of the NRL modified betatron gave a delay time of 34 μsec, which is less than 10% from the theoretical prediction of 37 μsec.