# Conservation of Energy: Comparison between momentum & magnetic fields

 P: 114 Conservation of Energy: Comparison between momentum & magnetic fields You have to look at the whole field interaction volume to get the total effect of the field interaction in electromagnetics, but individual points may be quite different to the sum effect. For example, if you look at two electrons near each other, then at the midpoint of the line joining them the energy density drops to zero as the fields exactly cancel each other out. For two electrons, there is a small region of electrostatic energy density reduction - the sphere whose diameter is the line joining the centres of the electrons - in the middle of a much larger and more significant region of increased energy density. Hence in the interaction between two electrons there is a small region of attractive forces (associated with the central reduced energy-density zone) that is entirely swamped by the repulsive forces. Do not think of the field lines as being distorted as the magnets come together. Rather, look at the composite field vector. If we imagine one of your magnets on top of the other, then in the horizontal plane that defines the midpoint between your magnets, all vertical field vector components exactly cancel out (vector addition). All that is left in that plane is the horizontal components, and at the centre of the plane there will be none. Essentially, in most static electromagnetic field interactions there are regions of both types - that is, where there is some cancellation of the electrostatic field (leading to a reduction in energy density and hence attractive forces) as well as regions where there is re-inforcement. It is relatively easy to perform the maths for electrons (which have simple polar fields), but I have not personally done it for magnets, since the geometry is highly variable. Nevertheless simple observation indicates a region of reduced energy density in the region between the magnets, swamped by a much larger region of repulsion elsewhere. P.S. If you want to look at the interaction between two electrons in more detail, start with the potential energy density function dU/ds = εE$_{1}$.E$_{2}$ where epsilon times the dot product of the electric field vector from one electron at a point and that from the other at the same point, is the potential energy density at that point. Integrating this function over all space gives the potential energy U=q$^{2}$/4∏εr and differentiating this with respect to separation gives the force F=-q$^{2}$/4∏εr$^{2}$ You can perform a similar integration for the magnetic potential energy between two magnets, but you have multiple geometries to choose from.