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The potential magnetic energy of a magnet or magnetic moment in a magnetic field is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to: = The mechanical work takes the form of a torque : = = which will act to "realign" the magnetic dipole with the magnetic field.
Magnetic potential energy is the form of energy related not only to the distance between magnetic materials, but also to the orientation, or alignment, of those materials within the field. For example, the needle of a compass has the lowest magnetic potential energy when it is aligned with the north and south poles of the Earth's magnetic field.
The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (magnetic pole strength), and the vector separating them. The magnetic dipole moment m is related to the fictitious poles as [ 7 ] m = p ℓ . {\displaystyle \mathbf {m} =p\,\mathrm {\boldsymbol {\ell }} \,.}
Magnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. Roughly speaking, the magnetic field of a dipole goes as the inverse cube of the distance, and the force of its magnetic field on another dipole goes as the first derivative of the magnetic field. It follows that ...
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
If the magnet is aligned with the magnetic field, corresponding to two magnets oriented in the same direction near the poles, then it will be drawn into the larger magnetic field. If it is oppositely aligned, such as the case of two magnets with like poles facing each other, then the magnet will be repelled from the region of higher magnetic field.
Magnetic charges = never occur isolated as magnetic monopoles, but only within dipoles and in magnets with a total magnetic charge sum of zero. The energy of a localized magnetic charge q m in a magnetic scalar potential is =, and of a magnetic charge density distribution ρ m in space =, where µ 0 is the vacuum permeability.
The outer potential U out must also be regular at infinity: both |r U| and |r 2 U| must be bounded as r goes to infinity. This ensures that the magnetic energy is finite. [10] Sufficiently far away, the magnetic field looks like the field of a magnetic dipole with the same moment as the finite body.