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Magnetic body force on the medium

How hard does the coil's fringe pull on the magnetised medium?

The formula
fm=μ0⁢M⁢G,Δp=fm⁢ℓg
fm
Body force per unit volume, N/m³
M
Medium magnetisation, A/m
G
Field gradient ∇H, A/m²
Δp
Pressure across the fringe, Pa
ℓg
Length of the gradient region, m
LaTeX 
fm = \mu_0 \cdot M \cdot G, \qquad Δp = fm \cdot ℓg

Work it out

kA/m

The magnetisation of the medium as a whole, per unit volume of slurry — for a 5 % micron-iron slurry in a 50 kA/m field, about 7.5 kA/m. Negative for a diamagnetic medium.

A/m²

How fast H changes along the tube at the coil's end, ∇H. A 50 kA/m field falling off over 25 mm is 2 × 10⁶ A/m².

cm

How far along the tube the gradient extends — roughly the coil's radius at either end.

Compare with a variation

Result

fm Body force 18850 N/m³

Force per unit volume of medium, toward the stronger field. Water's weight is about 9800 N/m³ for comparison.

Δp Pressure across the fringe 471.2 Pa

The body force integrated over the gradient region: the pressure difference one coil end can set up. Compare with the loop pressure drop.

With your numbers
18850 N/m³=μ0⁢7.5 kA/m⁢2000000 A/m²,471.2 Pa=18850 N/m³⁢25 mm
LaTeX 
18850\,\mathrm{N/m³} = \mu_0 \cdot 7.5\,\mathrm{kA/m} \cdot 2000000\,\mathrm{A/m²}, \qquad 471.2\,\mathrm{Pa} = 18850\,\mathrm{N/m³} \cdot 25\,\mathrm{mm}

Worth knowing

  • M itself depends on H inside the gradient (M = χ·H), so this is the force at the stated magnetisation, not a self-consistent one: across the fringe M falls with H and the true mean force is lower.
  • Across 2.5 cm of fringe this comes to 471.2 Pa — 48.14 mm of water column. Set that against the loop's pressure drop: a coil end pumps only if it exceeds the loss round the loop.

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Body force against field gradient Field gradient swept from 1000000 A/m² to 3000000 A/m² with everything else held at your numbers. The dashed lines cross where you are.
Body force against field gradientBody force rises from 9425 N/m³ to 28274 N/m³ as field gradient rises from 1000000 A/m² to 3000000 A/m². At your field gradient of 2000000 A/m² it is 18850 N/m³.50001000015000200002500030000100000015000002000000250000030000002000000 A/m²18850 N/m³Field gradient (A/m²)Body force (N/m³)
The formula behind the curve
fm=μ0⁢M⁢G,Δp=fm⁢ℓg
fm
Body force per unit volume, N/m³
M
Medium magnetisation, A/m
G
Field gradient ∇H, A/m²
Δp
Pressure across the fringe, Pa
ℓg
Length of the gradient region, m
LaTeX 
fm = \mu_0 \cdot M \cdot G, \qquad Δp = fm \cdot ℓg
What moves the answer Each input moved 10% either way, with the others held still, and the effect on body force.
What moves the answerBody force is most sensitive to Field gradient, which moves it by about 10% for a 10% change. It is least sensitive to Length of the gradient region, at about 0%.Change in the answer when each input moves by 10%-20%-10%10%20%Field gradient±10Medium magnetisation±10Length of the gradient region±0
The formula behind the curve
fm=μ0⁢M⁢G,Δp=fm⁢ℓg
fm
Body force per unit volume, N/m³
M
Medium magnetisation, A/m
G
Field gradient ∇H, A/m²
Δp
Pressure across the fringe, Pa
ℓg
Length of the gradient region, m
LaTeX 
fm = \mu_0 \cdot M \cdot G, \qquad Δp = fm \cdot ℓg

Method

  1. Convert the magnetisation to A/m and the gradient region to metres.
  2. The Kelvin force on a magnetised volume is μ₀·M·∇H: the magnetisation times the gradient of the field, times the permeability of free space (4π × 10⁻⁷ H/m). Its direction is toward the stronger field for a paramagnetic or ferromagnetic medium, away from it for a diamagnetic one.
  3. A force per unit volume acting over a length is a pressure. Multiply by the length of the gradient region for the pressure difference the fringe can sustain.

Assumptions

  • M is taken as given and uniform across the gradient region. In reality M is induced by the very field whose gradient is doing the pulling — M = χ·H — so it falls as H does across the fringe, and the force here is the force at the stated M, an upper figure for the region.
  • The gradient is along the tube. The radial gradient at a coil's end pulls the medium toward the wall as well, which is a mixing and a settling question rather than a pumping one.
  • The medium is a continuum. For a dilute slurry the force acts on the particles and they drag the carrier with them through viscous coupling; at micron sizes that coupling is tight and the continuum picture holds.
  • Nothing here is Meyer's. The Kelvin force is nineteenth-century magnetostatics; the magnetisation and gradient are the reader's, and the estate specifies neither.