Equation for the motion of an iron particle in a magnetic field

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SUMMARY

The motion of an iron particle in a magnetic field is governed by the principles of magnetic field density and force vectors. When placed between the poles of a stick-form magnet, the particle is attracted towards the area of highest field density, resulting in a trajectory that is not necessarily the shortest path. If the particle is positioned slightly off-center, it experiences a non-vertical force, causing it to approach the magnet at an angle rather than directly. The mathematical equation that describes this motion involves analyzing the magnetic field gradient and the resultant force vectors acting on the particle.

PREREQUISITES
  • Understanding of magnetic field theory
  • Familiarity with vector calculus
  • Knowledge of particle dynamics in physics
  • Basic concepts of magnetic field density
NEXT STEPS
  • Research the mathematical modeling of magnetic fields using Maxwell's equations
  • Study the Lorentz force law and its application to charged particles
  • Explore the concept of magnetic field gradients and their effects on particle motion
  • Learn about trajectory analysis in physics, particularly in non-uniform fields
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Physicists, engineering students, and anyone interested in the dynamics of particles in magnetic fields will benefit from this discussion.

ddddd28
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Hello,
Let's assume we put a small iron particle in a magnetic field. What is the equation of the route that the particle performs? Is it the minimal route length that is possible? (the magnet is a stick-form)
upload_2017-2-1_17-37-49.png
 
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It might look that way, but I don't think so.

As a general rule, the particle will experience an attraction in the direction of the biggest field density increase. That however means for example, if you place it right in the middle between the poles but above the bar, all it will do is go straight towards the middle of the bar. But, if you place it slightly say to the right of that point, the direction of attraction will sightly point towards to the right because the field lines also get more dense towards that direction.

So, in essence, slightly off-center the particle experiences a non-vertical force, which will make it hit the bar at an angle. That then obviously is not the direct route.
 
I wanted to know which mathematical equation describes the route
 

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