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Outside a narrow charged stream (say, a beam of ions or electrons) is the same as observing a current through a conducting wire - there is a circular magnetic field around it.

What would happen inside a charged stream (for example, inside a conducting wire or inside a solar flare)? I have a feeling that symmetry will rule that there is no magnetic field, but I am not sure.

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The mmf due to a current is determined by the current through the surface bounded by the closed path along which the magnetic field is integrated. A closed path within a cross-section of a conductor with, say, a uniform current density, will have a non-zero mmf associated with it and thus, a non-zero magnetic field exists within the conductor.

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The field is linearly proportional to r inside the stream (if the current density is uniform inside the beam) and falls off as 1/r outside. This is a simple application of Ampere's law. This leads to an attractive force compressing the beam, but always less than the electrostatic repulsive force pushing the beam out. The two balance when the beam is moving at the speed of light.

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