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
Best Answer
There is not really a simple answer to this question. It depends on the detailed material properties of the iron and the strength of the external magnetic field. What follows is a way to get an estimate of the force, but I would not be surprised if the results you get from this method are not terribly accurate.
When the iron is placed into a magnetic field, it gains a magnetic dipole moment $\vec{m}$ due to its magnetic susceptibility $\chi_m$. This is a property of the material of the ball, which tells you how strongly the atoms of the material respond to a magnetic field.
If the magnetic field is not too strong, and there are no hysteresis effects, then the resulting magnetic dipole moment of the ball is $$ \vec{m} = \chi_m V \vec{B} $$ where $V$ is the volume of the iron ball.
Finally, if the ball is sufficiently small (such that the strength of the coil's magnetic field does not vary substantially across it), then the force on the ball can be approximated as the force on a magnetic dipole: $$ \vec{F} = - (\vec{m} \cdot \vec{\nabla}) \vec{B} $$ which, after some mathematical manipulation, can be rewritten as $$ \vec{F} = - \vec{\nabla} \left(\frac{1}{2} \chi_m V B^2 \right). $$
So to calculate the force on the ball, you will need to know:
The volume of the ball. This is easy enough to figure out.
The magnetic susceptibility of the material of the ball. This is harder to determine; while the above-linked Wikipedia article gives a number for this, it is perhaps an over-simplification. To illustrate this, take a look at the table for the magnetic permeability of various materials; the permeability $\mu$ is related to the susceptibility by $\chi_m = \mu/\mu_0 - 1$. From this, we can see that permeability of the iron depends strongly on its purity, so this may be difficult to determine. (And I assume that when you say "iron" you mean "iron" rather than "steel"; steel is even harder to figure out.)
The magnetic field created by the coil as a function of position. This is also rather difficult to calculate from first principles, though it can be done (particularly if you're only interested in the case where the ball moves along the symmetry axis of the coil.) You could also in principle measure the magnetic field at several points along the desired path and interpolate between them to figure out the force. Also, remember that I said that the magnetic field had to be "not too strong" above; if the magnetic field is too strong, then the nice linear relationship between the magnetic dipole $\vec{m}$ and the magnetic field $\vec{B}$ breaks down.
Finally, note that the force will change as a function of position: closer to the magnet, $\vec{B}$ is larger and changes more rapidly with position, and so the ball will experience a greater force. This means that you'll have to use calculus to figure out the motion of the ball if that's the reason you want to find $\vec{F}$. But if you've gotten this far into this answer without getting lost, that shouldn't be terribly complicated for you.