Conservation of Energy: Comparison between momentum & magnetic fields

1. magnetics

32
Momentum and magnetic fields are both vector quantities.

If two bodies with the precise mass and speed collide head on (θ = 0), then momentum is conserved (they come to a complete stop) and energy is conserved (the kinetic energy is changed to other forms).

What then happens in the case of two identical strength magnetic fields that meet head on? A gauss meter placed precisely in between them would read zero if measured in the same direction of the field. Is the energy conserved simply by changing direction?

This is an oxymoron as energy is not a vector quantity.

Thank you.

2. pumila

114
When magnetic fields overlap and interact there is a change in the energy density of the field. This is what gives rise to forces when magnets are brought together, from F=dU/dl. Although the gauss meter reads zero there is actually an increase in the total magnetic energy in the system. This magnetic energy comes from the forces needed to push the magnets together.

3. magnetics

32
Thank you pumila,
I can see how forcing the magnets together might increase the total magnetic energy in the system. But I'm not sure about the gauss reading.
There may be a zero gauss reading in the direction of the local field vector, but wouldn't the field have been deflected in a perpendicular direction so that the magnitude of the vector does not change, but only the direction?
Thanks,

4. pumila

114
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.

Last edited: Mar 23, 2013