Electromagnetic inertia           Home / EM

Overview

Have you ever thought “If an electron induces a magnetic field when it moves, and magnetic fields contain energy, where does the energy to create this field come from?”

It comes from the force that accelerated the electron from rest - the work done in accelerating it (as the integral of the force over the distance) directly creates the energy of the induced field. Another way of saying this is that part of the electron’s inertia comes from supplying energy to the induced magnetic field.

We do not know the fine detail of the electron’s electric  field structure, only the way it falls off as the square of the distance when we are some way away from the electron (in terms of the electron size). So it is easier to deal with the inertia of simple electromagnetic fields. The equations are straightforward and derived for Newtonian velocities, using source equations to be found in any first-year physics textbook.

Electrostatic inertia - the basics

If a point in space has a vector electrostatic field of strength ‘X’ volts/meter, its energy density over space ‘dE_E/dS’ joules/meter³ is given by...

If this field is moving at a velocity vector ‘v’ relative to the observer it induces a vector magnetostatic field of induction ‘B’...

(Note that the symbol ‘x’ refers to the vector cross-product, so (X x v) is the vector X’s cross-product term with v, vectors all being in a bold typeface.)

This induced magnetostatic field has an energy density ‘dE_M/dS’ of...

The cross-product ‘(X x v)’ is zero where the vectors ‘X’ and ‘v’ are parallel to each other, and a maximum where they are orthogonal. At this maximum...

...so the ratio of the induced magnetostatic field energy density to the primary electrostatic field energy density is simply v²/c² at this maximum.

If the orientation between the electrostatic field and the velocity vector is other than the optimum of 90 degrees, the ratio of induced magnetic energy to the rest electrostatic energy is v².cos²/c². The “cos” term refers to the cosine of the angle between the velocity vector and the orientation of the primary electrostatic field lines.

Electrostatic inertia - a nightmare for Relativity Physics!
Click here to skip the heavy stuff!

Can we apply the maths to the electron on the assumption that the electron is a pure radial electric field?

Because of its spherical symmetry the electron can be dealt with by three orthogonal vectors - one along the direction of motion, and the other two normal to the direction of motion, with the rest energy of the electric equally distributed between them. The 1/3^rd along the direction of motion induces no magnetic energy, while the other two vector directions induce 1/3^rd of v²/c² each, so the total induced energy is 2/3^rd of v²/c².

Now the actual kinetic energy of an electron at Newtonian speeds is one half of v²/c², so this result is 4/3^rd of what we might expect. This is famous as the “4/3 Problem”. In 1922 Enrico Fermi, in the paper “Il Nuovo Cimento”, derived this result relativistically (the maths here is the Classical solution), but showed nothing new - after all the relativistic approach must reduce to the Classical approach at sub-relativistic speeds. Feynman’s “Lectures on Physics” showed how this 4/3 result violated relativity. The conclusion often made is that the electron could not be purely electromagnetic. However the calculation was made for a radial electrostatic field (albeit on the working assumption that the electron could be modelled by this) and hence the proper conclusion is that a radial electrostatic field violates relativity. By geometrical inference all electrostatic field structures violate relativity.

However, relativity problems pale into insignificance against a much greater problem. Suppose you accelerate a linear-geometry  field lying normal to the direction of travel from your rest frame into another one; you apply energy to it to accelerate it, giving it kinetic energy proportial to v²/c². Now suppose a mechanism in its new rest frame rotates the field to align it along the direction of travel; this takes no net work in that rest frame, but the induced field disappears together with all its kinetic energy. Where has the energy gone? Worse still, imagine a linear electric field that is moving with respect to you and simultaneously rotating on an axis normal to both the the direction of travel and the direction of the field- it will have a pulsating kinetic energy at a constant speed. What should we make of this apparent violation of the Conservation of Energy? We cannot suspect the laws of induction, which are well-proven - electric motors work because induced fields containing energy - but something is going on that our science has yet to encompass.

Maybe it comes from the fundamental assumption in science that the three spatial axis (and even the time axis) are  indistinguishable, but electrostatic induction has anisotropic behaviour that conflicts with this assumption.

All in all, motion-induced electromagnetic induction is a real nightmare for relativity physicists! - this despite the fact that it has been the well-understood workhorse of electric motors (such as hairdriers, trams, steel rolling mills) and generators (such as car alternators and power generating stations) for well over a century. There is an immense gulf between the scientists who work with relativistic particles and photons, and those who work with motion-induced induction; the theory of both is solidly proven by experiment and usage, but they are in conflict with each other.

Magnetostatic inertia

For completeness it is worth looking at the induction of a moving magnetostatic field.

If a point in space has a magnetostatic field of induction ‘B’, energy density...

which is moving at a velocity ‘v’ relative to the observer, it induces an electrostatic vector field of strength ‘X’...

 X = (B x v)

 ...with an energy density...

The induced field is zero when ‘B’ and ‘v’ are parallel, and reaches its maximum when they are orthogonal to each other. At this maximum...

...so again the ratio of the induced electrostatic field energy density to the primary magnetostatic field energy density is simply v²/c² at this maximum.

If the orientation between the magnetostatic field and the velocity vector is other than the optimum of 90 degrees, the ratio of induced electrostatic energy to the rest magnetostatic energy is v².cos²/c². The “cos” term refers to the cosine of the angle between the velocity vector and the orientation of the primary magnetostatic field lines.

Again, we have the same problem as with electrostatic inertia - the magnetostatic inertia is 4/3^rd that of ordinary matter. But there is another problem with partciles that have an intrinsic magnetic dipole - they should have directional inertia. Since the electron has the same inertia in all orientations, it follows that there is a discrepancy here to be resolved.

Conclusion

Electrostatic and magnetostatic fields exhibit inertial properties, but it is fair to say that there are issues awaiting resolution.

Appendix
Application to Inductance

The phenomenon of inductance is clear proof that the electric field of a moving electron has inertia.

Electrostatic inertia is associated with the concept of inductance, used in electric circuits; creating the magnetic field in an inductor takes energy that must be supplied by the external electromotive source. This energy is ‘L.I²/2’,  where ‘L’ is the inductance and ‘I’ is the current; the current is simply a count of the moving electrons, and hence is proportional to the total moving electrostatic field.

In order to make higher inductances the wires are formed in a loop or coil so that the induced magnetic fields from different electrons overlap and add. The electric fields from electrons in adjacent loops add linearly, so the energy involved increases as the square. This is why inductance ‘L’ is proportional to the square of the number of turns in the loop, and is clear proof that the electric field of the electron has inertia.

When current is flowing in a circuit through an inductor, and the circuit is suddenly broken open, a spark will jump across the break. This is caused by the augmented inertia associated with the electrons’ overlapping electric fields - they cannot stop dead, but have to dissipate their induced magnetic energy. This is a clear demonstration that an electron’s inertia owes much to its elecrostatic field.

Adding a ferrous core to such a coil increases the induced magnetic field dramatically, leading to much increased induced inertia.