Behavior of Conducting Solid or Liquid Jet Moving in Magnetic Field: 1) Paraxial; 2) Transverse; 3) Oblique

When a conductor moves through a nonuniform magnetic field, eddy currents flow that interact with the field to decelerate the conductor and perhaps change its trajectory, orientation and, if a liquid, shape. A rod of radius a = 1 cm and the density and electrical conductivity of melted gallium (γ = 6.1 g/cm, ρ = 26 μΩ cm) will decelerate 6.3 m/s in a 0.5 m ramp of paraxial field with a constant gradient g of 40 T/m (∆B = 20 T). The deceleration is proportional to ag∆B/γρ, independent of the velocity. The bar decelerates about twice as much in a 20 T, 0.5 m ramp of transverse field. A bar traveling at a shallow angle to such a field decelerates about 6.3 m/s. If the bar is 0.25 m long and moves at 20 m/s, it aligns with the field in ~10 ms, during which time it advances ~0.2 m. Assumptions: In all cases the conductor is a cylindrical rod or rectangular bar of length l and density γ . The electrical resistivity ρ is sufficiently high that the field generated by eddy currents never rivals the ambient magnetic field. Currents respond instantaneously to voltages, according to Ohm’s Law, without any inductive delay. Case (1): Rod Moving in Paraxial Field: 1a) Linear Ramp; 1b) Field of Long Coil In case (1) the conductor is a rod of radius a that is coaxial with an axisymmetric magnetic field and moves at a velocity v along their common z axis. Along the axis the field is purely axial: v B z B z z ( , ) ( ) $ 0 = . Zero divergence of v B implies that the radial component of field near the axis is, to first order, B r z r g z r ( , ) ( ) / = − 2 , where g z ( ) is the axial field gradient dB z dz ( ) / . A coaxial loop of radius r encircles a flux Φ( , ) r z of π r B z 2 ( ) ; the voltage induced around the loop is: V r z t d r z dt ( , , ) ( , ) = − Φ = −πr dB z dt 2 ( ) = −πr dB z dz dz dt 2 ( ) = −π r g z v t 2 ( ) ( ) . The voltage induces a circumferential current density: j r z t V r z t r ( , , ) ( , , ) = 2πρ = − r g z v t 2ρ ( ) ( ) . The current density interacts with the radial component of magnetic field to generate an axial force per unit volume: f r z t j r z t B r z z r ( , , ) ( , , ) ( , ) = − = −r g z v t 2 2 4 ( ) ( ) / ρ . Note that this is proportional to −v t ( ) , and so the force always is one of deceleration. The current density also interacts with the axial magnetic field to generate a radial force whose density is: f r z t j r z t B z r z ( , , ) ( , , ) ( ) = = −r g z B z v t z ( ) ( ) ( ) / 2ρ . In the paraxial case the magnitude of radial force density is much larger than the axial one, by the ratio B B z r / . This force compresses the conductor radially as it enters the field and tends radially to disperse a conducting jet as it exits. However, unless arcing maintains the circumferential path for current to flow, eddy currents should collapse, eliminating the forces on the jet—axial as well as radial. In any case, the radial boundary of the jet should remain within the confines of its enclosing tube of force. Any conductor whose boundary follows a tube of force sees no change in flux linkage, and hence develops no induced voltage, eddy current or force. Therefore, a liquid jet emerging from a field should expand no more rapidly than do the flux lines; its diameter should not double until the field has fallen by a factor of four. Integration of the axial force density gives the total force F z t ( , ) on the rod. Integrate with respect to r from 0 to a , and with respect to z from the rod’s trailing edge z z l − ≡ − to its leading edge z . Division by the mass of the rod, m a l = πγ 2 , gives the acceleration, dv dt / : F z t a G z G z v t ( , ) ( ) ( ) ( ) = − − − π ρ 4 8 ; dv dt a l G z G z v t = − − − 2