Magnetisation of a moving polarised dielectric

If the water is electrically polarised across the tube and moving, what magnetisation does that motion amount to?

M = Pe \cdot v, \qquad Pe = \varepsilon_0 \cdot (εr - 1) \cdot E

Method

1. Look up the medium's relative permittivity — 80 for water — and subtract one for the part that is the medium's own response rather than the vacuum's.
2. Multiply by ε₀ (8.85 × 10⁻¹² F/m) and by the electric field for the polarisation, in coulombs per square metre: the bound charge that appears on each face of a slab of the medium across the field.
3. A polarised medium in motion carries its bound charge with it, and a moving charge is a current. For velocities far below light, the magnetisation an observer at rest sees is P × v — its magnitude P·v when the flow is across the field, pointing across both.
4. Multiply by μ₀ for the flux density this magnetisation makes. Set it against the μ₀ × 7.5 kA/m of a 5 % iron slurry in a 50 kA/m drive field, and against the polarising field itself: ΔB ÷ E is (ε_r − 1)·v ÷ c², which is where the v/c lives.

Assumptions

• The low-velocity limit of the Minkowski relations for a moving medium, M = P × v to first order in v/c, with the flow perpendicular to the electric field. At 1.27 m/s the neglected terms are smaller by another factor of 10⁻⁸ and there is nothing to correct.
• A linear dielectric with the static permittivity. Water's 80 holds up to gigahertz; an oil's 2 is nearly all electronic and holds higher still.
• That the field can be sustained at all. A megavolt per metre is of the order of the breakdown field of ordinary water — distilled water holds a few times more, tap water less — so the polarisation shown is the most the medium could carry before it conducts instead. A good dielectric oil holds tens of megavolts per metre, but has a fortieth of the permittivity.
• No free charge. A conducting medium in a field carries a conduction current that swamps this, and is a different calculation.
• Nothing here is Meyer's. The archive is asked whether polarised water moving through a coil is, in effect, a magnet, and this is the answer standard electrodynamics gives: yes, to a part in 10⁹, which is the ratio of the pump's speed to light's.