***Contents*** Click any link below to view the article. You can return to this page by using your browser's back arrow. Electrodynamics Ideas you might not find discussed elsewhere, but all firmly grounded in Maxwell's Equations and the Lorentz Force Law. The more one works into the subject, the more one begins to appreciate the implications of these five equations. Oscillating Charge and the Emitted Radiant Energy per Cycle. In this article the radiant power emitted by an oscillating point charge is discussed. Two formulas are considered: (1) the LarmorLienard formula, and (2) the AbrahamLorentz formula. Both formulas are relativistically rigorous. Thus each applies to cases where the maximum speed of oscillation may range up to ~c. Consideration is limited to one dimension, where a point charge moves back and forth along the xaxis. The Electric Field "in front of" and "in back of" Two Point Charges. When 2 charges accelerate along the x axis, the electric field in front and in back of them is the same as in the case of 2 charges moving with constant velocity. SelfForces and Inertia. This article was featured on MySpace.com, in response to a thread on the origin of inertia. A Nonradiating(?) Oscillating Point Charge. When unequal charges oscillate in phase and one of them is approximately .076 of the other one, the smaller one neither absorbs nor emits energy. An Example of Field Energy Absorption. When unequal charges oscillate in phase, one of them may actually absorb energy emitted by the other one. The Equation of Motion for Multiple Charged Particles with Periodic Motion. Derives the extended form of Newton's 2nd law for such systems. Provides a different perspective on reaction forces and the driving agent's counteraction thereto. BiotSavart and Ampere's Law. Computes the magnetic field of an uncharged current loop, at points on another contour, and demonstrates the validity of Ampere's law. Briefly discusses how current loop interactions tie in with Newton's third law. Field Energy and Momentum. Analytically derives the energy and momentum in the fields of a spherical shell of charge that moves with a constant velocity. Shows that the kinetic energy resides in the magnetic field. A Derivation of the Electromagnetic Mass and SelfTorque of an Infinitely Long Solenoid. Does for an infinite solenoid what other articles suggest for spherical shells of charge. Analytic (no program)! On Oscillating Particles and the Forces that Drive Them. Comparison of the forces needed to drive uncharged and charged particles. Possible implications when electrons dive through atomic nuclei. Equations of Motion (Review). Pulls together many of the ideas about electromagnetic reaction forces, etc., discussed in other articles on this site. Orbital Precession in an Inverse Square Force Field. Demonstrates how the aphelion of an elliptic orbit precesses when the orbiting satellite's maximum speed is relativistic. The Electrodynamics of Length Contraction and Time Dilation. Demonstrates how Maxwell/Lorentz/Newton indicate that the shape of a moving Bohr atom model is length contracted, and how the period of the satellite's quasicycloidal path is time dilated. Suggests a generalization to all systems of interacting particles. A MaxwellBased Demonstration of the Radiation Reaction Force of Abraham and Lorentz. Specifies a motion for a point charge that rather irrefutably produces a "self" electric field right at the charge. Application of the Lorentz force law then produces the Radiation Reaction force first proposed by Abraham and Lorentz. On the NonRelativistic Electrodynamics of a Charged Spherical Shell. Discusses (a) the reaction forces when a tiny spherical shell of charge is accelerated, and (b) the counteracting, nonelectromagnetic agent force. Demonstrates how, in periodic motion, the net work per cycle done by the driving agent equates to the radiant energy flux per cycle into infinite space. Relativistic Reaction Forces. A relativistic continuation of "On the NonRelativistic Electrodynamics of a Charged Spherical Shell". Provides formulas for the relativistic inertial and radiation reaction forces. Includes Visual Basic programs that compute agent work per cycle, radiant energy flux per cycle, etc. A NonRadiating, Accelerating Charge. Derives an accelerated charge's motion that entails no radiation. Discusses the implications regarding the Larmor theorem. A Charged Particle on a Spring. A case where the inertial reaction force helps counteract the radiation reaction force some of the time. A logical derivation that oscillatory motion attenuates in time, with the emission of radiation. On a Needed Expansion of Newton's Second Law. Maxwell's equations and results derived by Abraham, Lorentz and others indicate that F=ma is not sufficient when F acts on a charged particle that moves periodically. The Equations of Motion for n Charged Particles With Periodic Motion. Derives the n forces, that must be applied to n charged particles that move in a given periodic way. Major feature: the driving agent's counteraction to the interactive (Lorentz) forces plays a key role. Driving Forces and Self Forces, Driving Powers and Field Energy Fluxes. Uses the field solutions for a point charge, at a spherical shell of charge's center, to compute the average fields in the shell's surface. Shows that the force required to make the charge oscillate equates to the negative of the force experienced by the shell in its own electric field, etc. Includes the first conclusive evidence that the Radiation Reaction force of Abraham and Lorentz is a real physical phenomenon. Accelerating Charges and Associated Inertial and Radiation Reaction Forces A demonstration that inertial and radiation reaction forces are associated with acceleration and the time rate of change of acceleration. Circling Point Charges. The net work per cycle, required to drive two diametrically opposed charges around a circle, is practically zero! That is, the tangential component of the interactive forces practically cancel out the radiation reaction force! Presumably the canceling out is perfect in the case of a spinning ring of charge. Fields and the Lorentz Transformations Twin Paradox Redux. Revisits the traveling and stayathome twins when the traveling twin is in a spaceship subjected to a constant thrust until it reaches a prespecified speed; then returns to a state of rest; then returns home. The Period of an Oscillating Clock. Shows how an oscillating clock's reading varies throughout a period of oscillation. A Derivation of the Lorentz Transformation of Space and Time Coordinates. Derives the transformations for x and t, from "first principles." The Reading of an Accelerating Clock. Shows how a clock slows down, with the passage of time, when it's accelerated. On the Relativity of Temperature. Inertial observers each have their own opinion about what the "true" mean squared speed of gas molecules is relative to the gas center of mass. The Transformation of Temperature. Systems cool off as they speed up! When F' = F = 0. When all of the components of an acting force sum to zero in one inertial frame, they must sum to zero in all inertial frames. On Required Classical Force Law Adjustments. In order to produce results consistent with the Lorentz transformation, classical force laws must be modified. The Electric Field at the Center of an Accelerating, Spherical Shell of Charge. The electric field inside a spherical shell of charge isn't always zero! Surprised? The Fields of a Translating, Uncharged Current Loop. Computes the electric and magnetic fields of loops that move perpendicular and parallel to their planes. Provides an alternate approach to translating current loop electric polarization. The Fields of an Infinitely Long, Uncharged Solenoid that Translates Parallel to it Longitudinal Axis. Long title, short article. Uses the general field transformations, as is done in "Current Loops ..." Current Loops That Move Parallel to Their Spin Axes. Specifies the fields of a moving current loop that has no velocity component lying in its plane. (Yes, there are no motioninduced electric dipole moments in this case.) A Caveat On Length Contraction. Moving systems, whose parts are fixed relative to one another in the system's rest frame, don't necessarily length contract when accelerated into another frame. How to Calculate the Electromagnetic Field in Frame K', All At a Single Moment. Given a knowledge of E(x,y,z) and B(x,y,z) in frame K at any moment t, provides formulas for E'(x',y',z') and B'(x',y',z') in frame K', all at the one instant t'. The Role of the Relativity of Current Loop Electric Polarization in the Electromagnetic Field Transformations. A more detailed discussion of why moving current loops are electrically polarized, and why this plays a role in the relativistic field transformations. Changing Rest Frames. K and K' perspectives when a particle accelerates out of initial rest frame K into final rest frame K'. Uses the Lorentz acceleration transformation. Very nonGalilean. Beyond Length Contraction. Rigid bars Might also bend when they move! The Curvature of Spinning, Translating Rods. Demonstrates how a rod, anchored by one end at the origin of inertial frame K, and spinning around the origin in the xyplane, is curved when viewed from inertial frame K'. On the Relativity of a Nonaccelerating Spring's Elasticity Constant. Demonstrates how the elasticity "constant" of a spring at rest in inertial frame K transforms when the spring is viewed from frame K'. A Curious Result in the Acceleration Transformation. A particle's component of acceleration might be zero in one frame, but nonzero in another. Learning to Love the Lorentz Transformation. A derivation of the Lorentz transformation. Great background reading. Translating Rods and SelfTorques. Discusses the interesting phenomenon that charges, on the ends of a translating rod, appear at first to exert an unbalanced torque on the bar. Radiation Two New Laws. All indications are, these are correct. On Oscillating Current Loop Polarization. Translationinduced electric dipole moments should result in dipole radiation. On the Independence of the Radiation Reaction Force from a Charge Distribution's Specifics. Suggests that the radiation emitted by a distribution of charge depends only on the motion of its "center of charge," and explores some ramifications. Also discusses the equivalence of accelerating distributions of charge and point charge substitutions. The Dual Roles of the Radiation Reaction Force. Sometimes its action is passed through to the driving agent, other times it acts on the driven charged particle. Suggested reading for planned future articles. On Negative Radiated Power. Discusses the idea of waveforms that propagate outward, while their energy fluxes inward. On the Larmor and AbrahamLorentz Formulas for Radiated Power. Discusses both the nonrelativistic and relativistic formulas for radiated power. Demonstrates the way the radiation from relativistic oscillators (wA~c) differs from that emitted by nonrelativistic oscillators. On the Fields of Radiant Pulses Emitted by a Relativistically Oscillating Point Charge. Computes and plots the electromagnetic field of a pulse of radiation, including what might be "seen" looking down the line of propagation of such a pulse. The Direction of Pulses Emitted by a Relativistically Oscillating Charge. The energy is "thrown forward" when things are relativistic. Driving Agent Power Expenditure, Oscillating Charge. Checks the spiked emission time derived in "Photons and Maxwell's Equations" by computing the agent power expenditure when an oscillating charge's maximum speed is relativistic. On Poynting Vector and Photon Velocity Directions. Happily, their directions transform identically! Interesting implications re light and virtual photons. The Stimulated Genesis of Radiant Energy from Electromagnetic Field Energy. Maxwell's equations and Energy Conservation together suggest a solution to a pair of paradoxes. Interesting implications regarding the "spontaneous" appearance of photons out of "empty space." On the Radiated Power of an Oscillating Charge. Discusses an interesting mechanical problem with the Larmor/Lienard formula for radiated power. LarmorLienard and AbrahamLorentz A Demonstration that When the Poynting Vector Points Away from an Oscillating Point Charge in Motional Rest Frame K, it May Point Toward the Charge in Frame K'. The author found this result, when first computed in Covariance of the AbrahamLorentz Formula for Radiated Power, Oscillation Along the xaxis, to be so counterintuitive that he ran the program for several other values of v (the speed of K' relative to K). (See AbrahamLorentz Plots of P'_{Rad} for Several Values of v to view the plots.) This article analytically demonstrates that this reversal in the direction of P_{Rad} is indeed an expected result. In doing so, it also suggests that, of the two formulas for radiated power (AbrahamLorentz and LarmorLienard), AbrahamLorentz is the one that appears to be consistent with the general field transformations. Covariance of LarmorLienard, Point Charge Going in a Circle in Motional Rest Frame K. As in straight line oscillations, the LarmorLienard formula for P_{Rad} works in both frames K and K'. Covariance of AbrahamLorentz, Point Charge Going in a Circle in Motional Rest Frame K. Uses the full, multiterm formula for the transformation of da_{x}/dt and da_{y}/dt. Flat line P_{Rad}(t), but oscillatory P_{Rad}'(t'). Covariance of the AbrahamLorentz Formula for Radiated Power, Charge Oscillating Along the yaxis. Same discussion as in "Covariance of the AbrahamLorentz Formula for Radiated Power, Oscillation Along the xaxis". Covariance of the LarmorLienard Formula for Radiated Power, Charge Oscillating Along yaxis. Same discussion as in "Covariance of the LarmorLienard Formula for Radiated Power, Oscillation Along the xaxis". AbrahamLorentz Plots of P'_{Rad} for Several Values of v. An addon to Covariance of the AbrahamLorentz Formula for Radiated Power. Demonstrates that the alteration of P_{Rad}(t), in K', is a relativistic effect that kicks in only at higher values of v. Covariance of the AbrahamLorentz Formula for Radiated Power, Oscillation Along the xaxis. AbrahamLorentz also works in all inertial frames. But the shape of the radiated power vs. time curve is markedly different in frame K', when the speed of K' relative to K is relativistic. Covariance of the LarmorLienard Formula for Radiated Power, Oscillation Along the xaxis. Like Newton 2, LarmorLienard works in all inertial frames. And the rate at which power is radiated is the same in all frames! Homopolar Generators and Motors SpinInduced E Fields and Angular Momentum. A curious bonus in the case of spinning, discshaped ceramic magnets. Throttling Mechanisms in Homopolar Motors. Both metallic magnet and ceramic magnet homopolar motors experience maximum angular rates. But the throttling mechanisms are distinctly different. A Suggested Difference Between Nonconducting and Conducting Discshaped Magnets. Nonconducting, spinning magnets have internal and external electric fields. Conducting magnets don't. On the Reaction Torque in a Homopolar Generator. Author's contribution to a very old stew. A Revised "Suggested Homopolar Motor. Eliminates one of the batteries in the original version. Minimizing Conduction Electron Heat Losses at the Peripheries of Spinning, Discshaped Magnets. Good news for OverUnity fans? On The Physics of Homopolar Generators and Motors. Discusses the forces/torques exerted upon a permanent magnet by a currentconducting wire, among other things. Provides a theoretical basis for symmetries discovered by GualaValverde and others. Electromagnetism and Rotational Relativity. The idea that the laws of electromagnetism work only in inertial frames, in the context of recent experiments with spinning, discshaped permanent magnets. A Suggested Method for Determining Whether the "Faraday emf" or SpinInduced Electric Field is the More Correct View in the Case of a Spinning Faraday Disc. The title says it all. But first read Faraday's "Paradox" and SpinInduced Electric Fields for background if necessary. On Over Unity Machines. Discusses why these machines may not be physically realizable. Faraday's "Paradox" and SpinInduced Electric Fields. Discusses an interesting difference between Faraday discs with closed circuits and test charges. A Suggested Theoretical Basis for Effects Measured by GualaValverde et al. Explains voltages measured by Jorge GualaValverde and others with spinning, discshaped permanent magnets. Provides insight into "homopolar generators," Faraday discs, etc. On the Dynamics of a Spinning, Discshaped Magnet. Discusses the dynamics of a modified homopolar generator. Specifies the torque that an external power source must deliver, in order to maintain a given current through the magnet/load circuit. A Suggested Homopolar Motor. A variation on the Homopolar Vehicle. This configuration doesn't roll along the floor, but rather acts as a source of mechanical power. A Suggested Homopolar Vehicle (Toy?). Dads take note. You can build this one with junior. Discshaped magnets can be purchased online. Quantum Theory Graphic models for the quantum wave function, and the "corpuscular" properties of the radiation emitted by oscillating charges whose maximum speeds are highly relativistic. Toward an Electrodynamically Stable Model of the Bohr Atom. Demonstrates that, given a nucleus in an appropriate circular orbit, the electron's radiation reaction force is counteracted by the interactive force of the nucleus upon the electron. The result is a system that does not radiate. On Measurement, Probabilities and "Subensembles". Discusses the quantum theory of measurement, which gives the author an uneasy feeling. Interfering Plane Y (Complex) Waves. Demonstrates the interference of waves all traveling in the same direction. A Suggested Generalization of Born. A curious blend of two seemingly exclusive views of Y. A Suggested Reinterpretation of Born. Interaction, not location, might be the right interpretation. Application of the "OR" Rule. Emphasizes the author's belief that the wave functions of virtually all ensembles of bound particles are not timeindependent. Time Evolution of YY*, 6 Energies, Square Well. Gives a more detailed look at the probability density as the resultant Y in an infinite square well propagates from the left to the right side of the well. On MultiEnergy Square Wells. Stationary states don't apply to square well ensembles populated by particles of multiple energies. On Quantum Wave Groups. Computes the probability density for a free electron using the spinning spiral model. On the Infinite Square Well Solution for Y. Good review article. Loss of Coherence and Bohr's Correspondence Principle. Debates what expression should be used for the deBroglie energy/frequency relation. 4 Quantum Mechanical Laws. A suggested follow up to Newton's Three. On the Spontaneous Genesis of Photons in an Expanding Universe. Suggests that the number of photons in today's universe may be much greater (and less energetic) than those in the big bang's initial fireball. The Fraction of Black Body Photons in (k,dk). Had he lived, Maxwell might have gotten the first curve that more or less resembled empirical reality. But Planck finally got it right. On the Temperature Independence of Energy Partitioning Among Photons in Black Body Radiation. The magic number appears to be 2.7. WHY??? A Suggested Parsing of Planck's Spectral Distribution Formula. The big news might be that there's 1 photon per unit volume per mode (or standing wave). The Evolution of a Quantum Paradigm. A historical review of the human mind's quest to make sense out of atomic physics. Includes qualitative comments on concepts discussed in the following two articles. Chatty. Enjoy! Simultaneous Energy States and NonRadiating and Radiating Atoms. Provides the conditions under which an atom, simultaneously occupying multiple energy states, will and will not radiate. Simultaneous Modes and NonRadiating Atoms. Atomic electrons occupy all accessible modes at once. That (so the argument goes) is why they don't radiate. "Quantum" Forces and Electric Motor Design. Uses Einstein's Work Function to suggest why electric motors can do work, notwithstanding the fact that magnetic forces cannot. The Effects of Dimensionality on a Ground State Hydrogen Atom Electron's Position Probability. The probability function depends upon how much is known about the electron's motion. Uses the Hydrogen atom ground state model first suggested in "A Stable, Maxwellian 2charge System." A Stable, Maxwellian 2charge System. Demonstrates how a nonradiating 2charge system can be theorized solely on the basis of Newton's laws and Maxwell's equation. Placed in this Quantum Theory section because the modeled particles have proton and electron parameters. Spinning Spirals and the Quantum Wave Function. A simple model for the quantum wave function. Makes Y less abstract. Leads into an analysis of the Finite Square Well. The Finite Square Well. Uses the Spinning Spiral model to analyze the finite square well. Has interesting solutions for the energy states. A Suggested Alternate Method for Normalizing the Quantum Wave Function. Demonstrates a paradoxical problem with the currently accepted method for normalizing Y. Suggests an alternate method which clears up the paradox, and may have interesting and farreaching ramifications in wave mechanics. The Master Magician. A discussion of some of the strange behaviors of elementary particles when they change energy states, etc. A Difficulty with the Magical Behavior of Elementary Particles. Discusses difficulties with the relativity of simultaneity, and the presently believed way that a particle transits standing wave function nodes. The Relativity of Current Loop Electric Polarization A topic that is rarely (if ever) discussed in physics texts is the electric polarization of moving current loops. It is a purely relativistic (Lorentz Transformation) effect which, once appreciated, enhances one's appreciation of the way a purely magnetic field in one inertial frame transforms to a combination of electric and magnetic fields in another. Experiments conducted by J. GualaValverde et al in Argentina seem to suggest rotational relativity ... an idea explicitly rejected by Feynman and others. Modeling a permanent magnet as an array of microscopic current loops, and applying the relativity of current loop electric polarization, may provide an alternate explanation of the experimental results. Revised Permanent Magnet Model. If permanent magnets do not have intrinsic angular momentum, then ... Current Loop Radial Reaction Forces. Shows the magnetic nature of Newton3 reaction forces when a current loop is made to go in a circular orbit at constant speed. On the Electromagnetic Inertia of Current Loops. Accelerationinduced magnetic field components act on the loop's legs to produce an inertial reaction force when the loop is accelerated. On Electromagnetic Angular Momentum and the Magnetic Field. Considers a different polarity for the discshaped magnet. Suggests that a spinning disc with one magnetic polarization has a net angular momentum, whereas one with a different polarization does not. A Simple Proof that Uncharged Current Loops are Electrically Polarized when in Motion. Introduction to the relativity of Electric Dipole Moments ... a phenomenon that appears to be of intrinsic importance in the electromagnetic field transformations. On the Difference Between Spinning Electromagnets and Spinning Permanent Magnets. Discusses why spinning, uncharged electromagnets have no spininduced electric fields, whereas spinning permanent magnets do. Provides a further theoretical basis for results obtained by GualaValverde et al, and suggests future experiments with electromagnets. Miscellaneous Fun Stuff Just what it says. Enjoy! Charge and the Equivalence Principle A timehonored conundrum which the author has received different explanations for from several experts. Velocity Anomalies. Why are orbiting satellites slowing down, Hmmm? On the Surface of a Whirlpool. The surface of a whirlpool, under appropriate conditions, is a parabola. Laurel and Hardy Discover Unexpected Symmetries. My favorite comedy team, ensconced in separate inertial frames, squabbles over who's right about length contraction, etc. Playing with Blocks. And you thought they were only for the small fry! On the Electric Polarization of Current Bearing Wires/Resistors. Charge density varies along a currentconducting wire/resistor. Stellar Aberration and Retarded Position. Aberrant or Retard ... it depends on your point of view. On Perceived Positions and Field Directions. Heirs of Tycho, Take Note? On the Perception of Moving Contours by a Single Eye. WYMINWYS. On Capacitor Earth's Magnetic Field. Earth's celebrated potential gradient engenders part of its magnetic field. Maxwell/Boltzmann and an Ideal Gas. Derives the gas pressure and a few other things from the Maxwell/Boltzmann distribution. Thoughts on Large Scale Electrical Energy Production. Not really all that fantastic. Suggested Proof of Fermat's Theorem. Quite brief, and very possibly at least part of what Fermat had in mind. Finding the Integer Solutions of a^{2}+b^{2}=c^{2}. Fermat no doubt knew this. The Amplification of Thermal Expansion/Contraction. Suggests how to join a number of bimetallic strips to produce a working "piston". Nuclear Forces and Interacting Current Loops. Looks at the hypothesis that the magnetic dipole moments of protons and neutrons may be caused by uncharged current loops. Computes the forces between such current loops for various magnetic dipole moment orientations and loop center separations. A Few Tests of Gauss' Law. Amazing results if you think about it! Gauss Tests, Relativistic. Tests of Gauss for a relativistically oscillating point charge; several surfaces. Spherical Surface, Ellipsoid, Egg. Bottom line: Gauss works over all of them, at all maximum oscillation speeds! Translating Solenoids, Maxwell's Aether and Electric Interactions. Just when you thought the aether was dead! Feynman's "Paradox" and Electromagnetic Moments of Inertia. Resolves a purported paradox suggested by the late R.P.Feynman. Also comments on Faraday's and Maxwell's views regarding electromagnetic inertia. Some Further Thoughts About Neutral Matter Interactions. More on imaginary analogues to the electromagnetic field. On the Constant Dimensions of Rotating Galaxies. Move over, Dark Matter? Analogies Between Heat and Radiation. Instructive parallels between a rigid sphere, driven in a viscous medium, and a spherical shell of charge driven in a vacuum, are discussed. When Gravity Balances the Lorentz Force. Discussion of a system where the repulsive force between two positive charges is equaled by the attractive force between two masses A "What If Magnetic Charges Existed" Model for a Stable Electron. A Stable Model for an electric charge with a magnetic dipole moment. Why the Total Energy of a Charge is Greater Than Its Field Energy. A puzzler which prompted Feynman to conclude that classical electromagnetic theory "ultimately falls on its face." (See Chapter 28 in The Feynman Lectures on Physics, V2 for a discussion.) This popular article demonstrates why Feynman was wrong. A Velocity Space Derivation of the Maxwell/Boltzmann Molecular Speed Distribution. Also popular. Inspired by a snippet from Maxwell's original paper. An excellent example of why working in "other" spaces can sometimes be the shortest route to a conclusion. How to Convert the Earth's Rotational Energy into Electric Power. Probably wrong, but a lot of fun to think about. Magnetic Bonding and Cold Fusion. A discussion of how two electrically neutral current loops might bond magnetically when in near proximity. The possible role of this process in the "cold" fusing of neutrons. Possible relevance to the nuclear force. The Zone. A fantasy episode in an elevator ... errr, a space pod. A Suggested Design Modification to Present, BlimpSupported Wind Power Systems. Counterrotating cylinders may solve some of the stability problems in blimpsupported winddriven generators. A Brief Overview of Particle Physics. Presents some relativistically correct cases of uncharged and charged particles. On the Gravitational Field of a Sphere of Charge. Develops the idea that the electric energy of a sphere of charge has a gravitational field. On the Counteraction of "Lorentz" Torques. Considers the interesting result that 2 charges, at rest in inertial frame K, experience a torque in inertial frame K'. Points out the importance of taking into account the mechanical restraint that holds the charges at rest relative to one another. On a Quantum Toy. A suggested macroscopic example of the Uncertainty Principle at Work. A Brief Overview of Particle Physics. Good background reading. On Charge/Charge Mass/Mass Interaction Analogs. Grounded in the ageold observation that charge and mass interactions are mutually inverse square forces.
