How to map out a true 3d image of the magnetic isopotential surfaces of a magnet. However, they're mistakenly calling the isopotential surfaces the magnetic field in that video (see attached image).
This is nothing new. It is just a different representation of the magnetic flux.
The linear force acting on that small magnet in the video is governed by the well-known formula for a force on a small magnetic dipole
m in magnetic field (magnetic flux density vector field
B):
F = ∇(m · B)In the magnetostatic case (no free currents,
∇ × B = 0), this simplifies to the equivalent directional derivative form:
F = (m · ∇)B.
This equivalence is the key - it means the entire problem reduces to computing the Jacobian tensor
∂Bᵢ/∂x at the dipole's location.
Each X,Y,Z component of the force acting on the dipole
m is simply one row of that tensor scaled by its dipole moment magnitude.
All of that is
directly derivable from the 3D flux map of the permanent magnet by the Jacobian transformation.
A line connecting points where the small magnet
m experiences the same linear mechanical force is the Isoforce line (or Isodynamic line). It is
not the Isopotential line !
Isopotential or Equipotential lines (surfaces in 3D) are not the same as Isoforce lines/surfaces !!!
The isopotentials of a dipole system are lines/surfaces where the potential energy
U = −m·B = const, and the force vector
F = −∇U is perpendicular to those lines/surfaces.
The isoforce lines/surfaces (
|∇U| = const) are a completely different family of lines/surfaces.
Using the word "isopotential" brings up the connotation of the magnetic vector potential
A, which is something very different from the linear force acting on a small magnet
m.
To get from
A to the force
F acting on
m you must:
1) Take the curl of
A to recover
B2) Form the dot product
m·B3) Take the gradient of that scalar
Here is the full hierarchy:
| Object | Symbolic repr. | What it is | SI Units |
| ---------------------------- | --------------------- | ---------------------------------- | ------------- |
| Magnetic vector potential | A | Potential field; B = ∇ × A | Tesla·meter |
| Magnetic flux density | B | Physical field; acts on dipoles | Tesla |
| Dipole potential energy | U = −m·B | Scalar energy of dipole in field | Joule |
| Force on dipole | F = −∇U = ∇(m·B) | Gradient of the energy | Newton |
| Space | s=vt | Prison for your mind | Meter |
What I have written above does not refute what Smudge has replied to you. It is a different, albeit a related subject.
His
paper actually addresses the magnetic energy inside the space occupied by ferromagnetic materials. My reply - does not.