Editing Potential energy (section)

=== Magnetic potential energy ===
The energy of a [[magnetic moment]] <math>\boldsymbol{\mu}</math> in an externally produced [[Magnetic field|magnetic B-field]] {{math|'''B'''}} has potential energy<ref>{{cite book|last=Aharoni|first=Amikam|title=Introduction to the theory of ferromagnetism| date=1996|publisher=Clarendon Pr.|location=Oxford|isbn=0-19-851791-2|edition=Repr.| url=https://archive.org/details/introductiontoth00ahar}}</ref>
<math display="block">U=-\boldsymbol{\mu}\cdot\mathbf{B}. </math>

The [[magnetization]] {{math|'''M'''}} in a field is
<math display="block"> U = -\frac{1}{2}\int \mathbf{M}\cdot\mathbf{B} \, dV, </math>
where the integral can be over all space or, equivalently, where {{math|'''M'''}} is nonzero.<ref>{{cite book| last=Jackson| first=John David| author-link=John David Jackson (physicist)|title=Classical electrodynamics| date=1975| publisher=Wiley| location=New York| isbn=0-471-43132-X| edition=2d|url=https://archive.org/details/classicalelectro00jack_0}}</ref>
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. If the needle is moved by an outside force, torque is exerted on the magnetic dipole of the needle by the Earth's magnetic field, causing it to move back into alignment. The magnetic potential energy of the needle is highest when its field is in the same direction as the Earth's magnetic field. Two magnets will have potential energy in relation to each other and the distance between them, but this also depends on their orientation. If the opposite poles are held apart, the potential energy will be higher the further they are apart and lower the closer they are. Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.<ref>{{cite book|first=James D.|last=Livingston|title=Rising Force: The Magic of Magnetic Levitation| publisher=[[President and Fellows of Harvard College]]|date=2011|page=152}}</ref><ref>{{cite book| first=Narinder| last=Kumar| title=Comprehensive Physics XII| publisher=Laxmi Publications|date=2004|page=713}}</ref>