Magnetic force driving a small iron ball

[Elementrist]

How can I calculate the force from magnetic field of a solenoid, grabbing a small iron ball?
I want to use the good old simple F=ma formula in order to calculate the ball's acceleration.
But can't find a formula to somehow convert the known field quality (in unit Tesla) to Force (in unit Newton).

These are known:

• The magnetic field of the solenoid in mT at the point where ball is placed initially.
• The magnetic field of the solenoid in mT at exactly the center of the coil.
• The inductance of the coil and its DC resistance.
• the mass of the iron ball.
• Initial speed of the ball (which is zero)

I found some formulas related to magnetic fields and force, searching for these keywords, but they contain q (electric charge) and B (flux) and other vector qualities I know nothing about or seem irrelevant to my question.

Please help me with this calculation, or at least guide me by giving me words I can search for.

 

[Vanadium 50]

Those equations that "seem irrelevant"? Those are exactly the ones you need.

Think about it this way: in an ideal solenoid where the field is constant everywhere, which way does the ball move?

 

[Elementrist]

I know it moves toward the point with the most flux intensity. (Or field intensity, which ever the right terminology is).
The problem is the force that forces that movement, is not known to me.
There are formulas that measure that force for a moving charge (q) but not for a solid stationary iron mass.

I asked this question on another forum and someone said it's not easy to calculate, since it's related to magnetic susceptibility of iron and also the distance from coil center (where the field or flux(?) is the strongest) is an important factor to consider.

 

[rude man]

I know it moves toward the point with the most flux intensity. (Or field intensity, which ever the right terminology is).
The problem is the force that forces that movement, is not known to me.
There are formulas that measure that force for a moving charge (q) but not for a solid stationary iron mass.

I asked this question on another forum and someone said it's not easy to calculate, since it's related to magnetic susceptibility of iron and also the distance from coil center (where the field or flux(?) is the strongest) is an important factor to consider.

You can calculate the axial B field outside the solenoid using Biot-Savart. This ignores the distortion of B due to the iron unfortunately, so assume a strong solenoid B field and a small ball!

You then need to make a simplification: susceptibility of the iron is infinite. This is a very good assumption for iron or other high-permeability material.

Principle: There is "suction" pressure at every point along the surface of the ball. This pressure is equal to the magnetic energy density at each point (can be derived from virtual work principle). Thus, the "suction" force at each differential area dA everywhere along the ball is $F = BH/2~ dA$.

However, this force is everywhere normal to the surface so B has to be the component of B normal to the surface, and the force itself is also of course normal. I'm thinking the normal component of B could be $B cos(\theta)$ with $\theta$ the angle between the normal and B which is assumed axial with the solenoid.

You then would need to do some amount of integration.

 

[hutchphd]

How can I calculate the force from magnetic field of a solenoid, grabbing a small iron ball?

Small compared to what?? It matters.