Force between 2 parallel magnetic dipole moments

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SUMMARY

The force of attraction between two parallel magnetic dipoles, both pointing to the right, is determined by their alignment and proximity. When two dipoles are aligned in the same direction, they exert an attractive force on each other due to the negative potential energy associated with their interaction. The energy is further minimized as the dipoles approach each other, resulting in a stronger attractive force. The relationship is quantified using the equations U = -m · B and F = -∇U, where U represents potential energy, m is the magnetic moment, and B is the magnetic field from the other dipole.

PREREQUISITES
  • Understanding of magnetic dipoles and their properties
  • Familiarity with potential energy in magnetic systems
  • Knowledge of vector calculus, specifically gradient operations
  • Basic principles of magnetic fields and forces
NEXT STEPS
  • Study the derivation of the magnetic dipole-dipole interaction formula
  • Learn about the implications of torque in magnetic systems
  • Explore the concept of magnetic field lines and their effects on dipoles
  • Investigate the role of distance in magnetic interactions and field strength
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Physics students, educators, and researchers interested in electromagnetism, particularly those focusing on magnetic dipole interactions and their applications in various fields.

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Homework Statement


Find the force of attraction between 2 magnetic dipoles a distance r apart. Both dipoles point to the right.

Homework Equations

The Attempt at a Solution


All I need help with is figuring out how to determine if the force is attractive or repulsive between the 2 dipole moments. From the question, it seems as though I can conclude that 2 magnetic dipoles pointing in the same direction attract each other. But I need a more fundamental way to figure this out. If I'm given 2 dipoles a distance r apart (where r is not huge) and with some orientation (relative to each other), how do I determine whether there is an attractive force or a repulsive force?
 
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Hello again. ## U=- m \cdot B ## where ## B ## is the field from the other dipole (magnetic moment). ## F=- \nabla U ##. (One thing that isn't completely clear from the statement of the problem=Presumably the dipoles are pointing along the x-axis and are a distance r apart on the x-axis.) ## \\ ## The magnetic field from both magnetic moments points from left to right (surrounding the magnetic moment), and both magnetic moments will thereby be aligned with the field from the other magnetic moment, making the energy negative for each. The energy becomes even more negative if the dipoles get closer together because the field that it feels from the other dipole will be stronger. The system will tend to go to the state of lower energy=thereby the force is attractive. (It should be noted the reason ## U=-m \cdot B ## (with a ## cos(\theta) ##) is because the torque ## \tau=m \times B ## (with a ## sin(\theta)) ## and ## U=\int \tau \, d \theta ##.)
 
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