Helical electromagnetic waves        modified 20110804
We simplify and imagine nature to be composed of machines which we can understand. The cosmos is a machine. We use these machines by analogy to explain the behavior of nature. We are very far from understanding nature. The planetary atom and the ring electron are examples of tiny machines which illuminate matter. We attempt to extend the realm of the tiny machines to the electromagnetic wave and to the flux of energy in the electromagnetic wave. Figure 1: They only look like bird wings. The red and green waves are perpendicular and in phase sine waves. They come to a zero point on the line at the same place and time. The energy at this time is discontinuous. J.C. Maxwell used the elastic medium of the luminiferous aether to store the energy of the waves like a compressed spring for the next cycle. Hertz concurred. Three problems: no aether, no spring, no obvious mechanism to possess or transfer energy. This is the oddly usual view of electromagnetic waves. Figure 2: Click image to animate! We choose a slightly different path. The red and green waves are perpendicular but they are out of phase by 90 degrees. They are sine and cosine waves. When one wave has maximum energy the other wave has minimum energy. The total energy is continuous and conserved. Pendulums, Hook's law oscillators, current and voltage in LC oscillators and transmitter antennas all share this relationship. Light has the same behavior. All share the same math. Figure 2A: Click image to animate! This is like figure 2 but these are stylized sine and cosine waves. We see them as flux tubes with their thickness proportional to their energy content. Their zero crossings are along the line. Each plasticine element of the flux tube begins and ends at a point. We see a continuous cycle of red up, green left, red down, green right, red up. This is something like a Halbach array. Energy alternates between the red and green waves as each wave induces the other through its own collapse. The maximum of energy moves forward and rotates. Light has spin and can transfer angular momentum. Figure 3: On the left, Faraday's law: The green flux of B times the area of the loop equals red -E times the circumference. A flux causes a looping around or a looping around causes a flux.
On the right, Ampere's law plus Maxwell: The red flux of E times the area of the loop equals green B times the circumference. A flux causes a looping around or a looping around causes a flux.
Galaxies and solar systems may form around currents in space as in the plasma universe. This is a flux causing a looping around. The disks around proto-stars frequently have jets as do galaxies. This is a looping around causes a flux (jet). Faraday's and Ampere's laws at a huge scale. Figure 3A: This is a chain using the laws of Faraday and Ampere from figure 3. If these are standing waves they might be self sustaining if the chain were looped into a ring. Figure 3B can also be looped into a ring as in figure 5. These rings could be related to Bostick's plasmoid or these rings flipping end over end might look like a sphere and could be related to ball lightning, articles by Louis or Wiki. Figure 3B: The four strands of figure 3A have been divided into two halves which now follow the sine and cosine waves of figures 2. Each half is identical but they travel in opposite directions. Figure 2B: This is another view of figure 2. The red and green flux tubes are still perpendicular but they are drawn flat so they can be scaled. Above the waves is an induction indicator showing how the waves are transformed.

Working sequentially left to right showing cause and effect for the four steps of one wavelength of figures 2 while using figure 3:
• The red in equals the green out. Ampere's law.
• The green in equals the red out. Faraday's law.
• The red in equals the green out. Ampere's law.
• The green in equals the red out. Faraday's law.

The speed of light: is the speed that the green B magnetic field or the red E electric field, loops around and augers to the right. The index of refraction, of the medium, reduces the velocity. The quarter wavelength pulse of flux, through the quarter circle of area, produces a quarter wavelength looping around the circumference, a quarter wavelength advance of the electromagnetic wave, in wavelength/(4*c) seconds. Top view, top: Red exerts a torque around the green axis, which is into the page, in the top view.
Side view, bottom: Green exerts a torque around the red axis, which is into the page, in the side view.
These waveforms are not the right shape. They do not go to zero at a change in direction. They are fun to make out of colored wires, see Appendix 5. The red and green are correctly, 90 degrees out of phase. They demonstate the torque around a perpendicular axis. A torque around two perpendicular axises produces a torque perpendicular to both in a gyroscope. Here the axises of the torque is moving with the waves. How does this work? This is gyroscopic precession and movement in the direction of travel of the wave. This is a very odd gyroscope indeed. We will explore this later using Euler's equations.

Pump analogy:
The flux of E or B through an area makes a 90 degree turn and then B or E rotates around the circumference. In a centrifugal pump, the liquid enters the eye of the pump, the center of the impeller, makes a 90 degree turn and is slung outward by the motor along a path that ends with rotation along the circumference.   Figure 2C, Helical electromagnetic waves: This is an end view of approaching light beams of figures 2. The peaks of the cross sections through the flux tubes are shown as colored circles. The peaks are also shown on figure 2B. They are marked by the verticle lines through the flux tubes. It is easy to follow a clockwise path around figure 2c starting at the top, +E→+B→-E→-B→+E or with the red sine→green cosine→red -sine→green -cosine→red sine. Light and all electromagnetic radiation can be seen as a sequential machine advancing on a helical path with four fluid transitions per wavelength. The helical structure and the transition delay between the peaks accounts for the velocity and refraction of light. The transition delays are proportional to the wavelength of the light and the medium. Each transition advances the light like a screw by a quarter wavelength. The beam having opposite polarity on opposite sides would be rotated in a magnetic or electrical field. A verticle slit would show only red and a horizontal slit would show only green. This is polarization. Light can go through a hole smaller than its wavelength. Thomas Young's double slit experiment and diffraction at an edge are explained by this structure. Use your thumb to hide the green of figure 2C, the beam is a double source with electric field interference. Use your thumb to hide the red, the beam is a double source with magnetic field interference. If you use your thumb is it still a Gedankenexperiment? This view of nature has some utility. What kinds of devices does it make possible?

This question was answered. By adjusting the phase of parallel beams of light, rotating the polarization, the beams may be made to attract or repell each other. This is demonstrated in this Nature article or this Discover article.

Do we pretty much know everything? We have no deep knowledge of the underlying reality. Our ignorance is more profound than just the missing link between quantum mechanics and relativity. Perhaps both are flawed and the pieces that each contributes to the puzzle of reality need to be rearranged by cherry-picking the best pieces of each. Science is a puzzle. When you close your eyes or walk in the dark to another room, you still know where you are because of your internal world model. Consciousness is seeing oneself mirror-like in ones internal world model. Our knowledge of reality is as riddled with voids as is swiss cheese. Our consciousness papers over these voids so our world model is perceived as smooth and uncomplicated for quick actions necessary for survival. We evolved to quickly see the face of a predator or prey in the shadows. We also see faces in the clouds. Some see the face of God. Apparently, we only see the shiny bits not the voids. We assemble these bits to quickly paint a shallow picture of reality and proceed to live our lives by rules-of-thumb based on these perceptions.

What are forces? We see the puppets move but we don't see the strings. How are forces propagated? What is their velocity of propagation? What is inertia? These are big unanswered questions which are close to home. Do you see the absurdity of trying to answer deep question about reality, like these, with sentences which include words like "virtual"? Do you see virtual photons in the clouds? Are you seduced by these shiny bits? Do our perceptions and behaviors seem so-ape-like? Right hand rule: When you grab a ring with your right hand, the thumb points in the toroidal direction along the ring while the fingers curl through the area of the ring in the perpendicular poloidal direction around the tube. Poloidal flux, which occurs through an area, and perpendicular toroidal looping around, which occurs along a circumference, are always associated. A flux or current has magnitude and direction. It has a rate of change d/dt, if its direction changes, even if its magnitude, area or circumference do not change. The flux of red E and green B, seen below, are as real as a mudslide. If they are real, what are they? We know their units. The mathematical interaction of units tells us something about the consistancy of our world model but one might mistake superficial knowledge of units for the deep knowledge of reality. Figure 3: Ampere's and Faraday's laws:
The red E electric field has units of volts/meter, kg*m/(A*s3) = kg*m/(s2) *1/(A*s),
E = force/q, force per charge or force per (amp*second).
force = q*E. q = A*s is charge.

The green B magnetic field has units of Weber's/m2 = Teslas = kg/(A*s2) = kg*m/(s2) *1/(A*m) ,
B = force/(charge*velocity), charge*velocity = amps*seconds*meters/second = amps*meters.
force = B*q*v = B*A*s*m/s = B*A*m. v is velocity. This B*q*v force is what keeps charge in orbit in the electromagnetic wave, cyclotron, tokamak and ring electron.
B*q*v = kg*v2/r, where kg*v2/r = centrifugal force. r is the orbital radius.
B*q*v acts like a central force to keep the charge and its energy equivalent of mass in orbit. We are comfortable with the not-so-obvious central force of gravity holding the planets in orbit. This central force is more obscure. There is no central object to provide a central force. There is only B*q*v. The direction of the moving charge is changed by B. How does it work? We know how it acts sometimes. We seek a metaphor to describe this peculiar force.

The above two forces are equal in an electromagnetic wave where the velocity = c.
q*E = q*c*B, or
E = c*B, in an electromagnetic wave.

Poincar� stress and energy density: �Poincar� stress� has to be present to prevent the electric charge of an electron from flying apart due to the Coulomb repulsion". See Paul Marmet.
q*E = q*c*B: The electrostatic force of repulsion of the charge equals the magnetic pinch force of attraction of the charge when the charge moves at the speed of light.
E = c*B, cancelled q, units are volts per meter, force per charge or kg*m/(A*s3)
E2 = B2*c2, squared.
E2 = B2 *1/(u0*e0), substituted c2 = 1/(u0*e0)
E2*e0 = B2/u0 = 1.55E25_kg/(m*s2), The energy density or pressure of the E and B fields are equal.
force/area = energy/volume = kg/(m*s2) = pressure = Pascals
kg*m/s2 /m2 = kg*m2/s2 /m3 = kg/(m*s2)

This is the magnetic pinch pressure equals the electrostatic pressure of repulsion. This magnetic pinch pressure restrains the charge to the thin ring of the electron like a hose restrains water.

Converting Ampere's law to Faraday's law/(c*u0):
d(E*pi*r2)/dt *e0 = 2*pi*r*B/u0 = amps. Start with Ampere's law. c*u0 = 376.73_ohms.
d(B*c*pi*r2)/dt *1/(c2*u0) = 2*pi*r*E/(c*u0) = amps. Substitute E=B*c, e0=1/(c2*u0), B=E/c.
d(B*pi*r2)/dt *1/(c*u0) = 2*pi*r*E/(c*u0) = amps, collect terms, Faraday's law/(c*u0). • Figure 3: First on the left, Faraday's law/(c*u0):
d(B*pi*r2)/dt /(c*u0) = 2*pi*r*E/(c*u0) = amps.
The poloidal green flux of B/(c*u0) times the area of the ring equals the toroidal red E/(c*u0) times the circumference of the ring. Red exerts a torque around green. B, E and the area of their loops may be constant but there is looping. A change in direction is also a change over time in the flux, a dB/dt or dE/dt.
• Second: The poloidal green flux which was shown as a green arrow is now shown as a green poloidal looping around the red torroidal current. The green flux is still out of the ring like the north pole of a magnet.
• Third, Ampere's law:
e0*d(E*pi*r2)/dt = 2*pi*r*B/u0 = amps. Maxwell's changing red poloidal displacement current times the area of the tube equals the toroidal green current times the circumference of the tube.
Green exerts a torque around red. This is a cross section through the second figure. It shows a single green loop of the poloidal flux around the tube and a piece of the red ring is now shown as a red arrow. In this cross section, the former red toroidal is now red poroidal and the former green poroidal is now green toroidal. This perpendicular transformation changes our viewpoint from Faraday to Ampere.
• Forth: The poloidal red flux which was shown as a red arrow is now shown as a red poloidal looping around a green torroidal current. The red flux is still out of the ring. A changing magnetic flux through a circular area generates a loop electric field which accelerates the electrons in a Betatron. Faraday's law and the electromagnetic wave, +B→-E or -B→+E:
E and B are sine and cosine waves because they are ninety degrees out of phase. B is the cosine since it has a sign change in its derivative. Lenz's law comes from the sign change in the derivative. d(cos)/dt = -sin or d(-cos)/dt = sin. Faraday's law is applied twice per wavelength so there is no net sign change per wavelength since, -1*-1 = 1. This sign change does not occur in Ampere's law, noting d(sin)/dt = cos or d(-sin)/dt = -cos, does not have a sign change.
Our derivation of Faraday's law starts with the idea that the rate of change of B is 4*pi*B times the frequency of the wave.
d(B)/dt = 4*pi*B*frequency, kg/(A*s3) Teslas/second
d(B)/dt = 4*pi*B *c/(2*pi*r), frequency = c/wavelength = c/(2*pi*r). Frequency measures how many loops something, moving at c, does in the ring 2*pi*r per second.
d(B)/dt = 4*pi*-E/(2*pi*r), B*c = -E, in an electromagnetic wave. See appendix 2.
2*-E = r*d(B)/dt, group terms, volts/meter = kg*m/(A*s3) = kg*m/s2 * 1/(A*s) = force/charge
2*pi*r*-E = pi*r2*d(B)/dt, multiply by (pi*r)
2*pi*r*-E = d(B*pi*r2)/dt, or -E*ds = d( B)/dt, Faraday's law. volts = kg*m2/(A*s3) = amps*resistance = energy/charge = watts/amps. Toroidal red -E times the circumference of the loop equals the rate of change of the poloidal magnetic flux of green B times the area of the loop. Faraday's law.
When we divide the voltage on both sides of Faraday's law by the resistance of a loop or coil of wire then we get Ohm's law: volts/resistance = amps.
2*pi*r*-E /resistance = (d(B*pi*r2)/dt) /resistance, amps or E*ds / resistance = d( B)/dt / resistance, amps. The toroidal amps in the loop equals the poloidal flux of amps through the area of the loop. This is the reversing current seen on a galvanometer, when hooked to a coil of wire, while a magnet is inserted and removed from the coil of wire. This classic experiment is strong direct evidence for Faraday's law. E*ds = -d( B)/dt,
integral form of Faraday's law. Hyperphysics or Wiki E*ds = 2*pi*r*-E, the line integral of the electric field equals the circumference of the loop times E.
d( B)/dt = d(B*pi*r2)/dt, the rate of change of the magnetic flux equals the rate of change of B times the area of the loop. A changing electric flux through the circular area of a plate capacitor generates a loop magnetic field. Ampere's law and the electromagnetic wave, E→B:
Our derivation of Ampere's law is very similar to Faraday's law. The rate of change of E is 4*pi*E times the frequency of the wave.
d(E)/dt = 4*pi*E*frequency, kg*m/(A*s4) volts/(meter*seconds)
d(E)/dt = 4*pi*B*c*frequency, E = B*c, in an electromagnetic wave. See appendix 2.
d(E)/dt = 4*pi*B*c*c/(2*pi*r), frequency = c/wavelength = c/(2*pi*r). Frequency measures how many loops something, moving at c, does in the ring 2*pi*r per second.
2*B = r/c2*d(E)/dt, kg/(A*s2) Teslas, collected terms
2*pi*r*B = pi*r2/c2*d(E)/dt, multiply by (pi*r)
2*pi*r*B = 1/c2*d(E*pi*r2)/dt, or B*ds = 1/c2*d( E)/dt, Ampere's law,

B = kg*m/(A*s2) = kg*m/(s2) * s/m * 1/(A*s), force per velocity per charge or force/(amps*meters).
Force = B*q*v = B*A*s*m/s = B*A*m. v is velocity.
Toroidal green B times the circumference of the loop equals one over c squared times the rate of change of the poloidal electrical flux of red E times the area of the loop. Ampere's law.
2*pi*r*B = e0*u0*d(E*pi*r2)/dt, or B*ds = e0*u0*d( E)/dt, Ampere's law. 1/c2 = e0*u0.
2*pi*r*B/u0 = e0*d(E*pi*r2)/dt, amps or 1/u0* B*ds = e0*d( E)/dt, amps. The left hand side of the equation, the toroidal amps, due to B, in the loop equals the right hand side of the equation, the poloidal flux of amps, due to E, through the area of the loop. The right hand side of the equations are Maxwell's displacement current. We will use this, amps equals amps form, along the wavefront. This is the form of Ampere's law used in the Ring Electron.

Integrals of Ampere's law: B*ds = 1/c2*d( E)/dt B*ds = 2*pi*r*B, The line integral around the curve equals the circumference of the loop times B.
d( E)/dt = d(E*pi*r2)/dt, The rate of change of the electric flux equals the rate of change of E times the area of the loop. B*ds = I*u0 + 1/c2*d( E*da)/dt. I is amps. Maxwell's modification of Ampere's law per Hyperphysics or per Wiki B*ds = I*u0 + e0*u0*d( E)/dt. Substituted e0*u0 = 1/c2. The above is written more clearly.
1/u0* B*ds = I + e0*d( E)/dt. Divided by u0.
2*pi*r*B/u0 = I + e0*d(E*pi*r2)/dt = amps. These equations are written much more clearly without the integral and flux symbols. This makes them more accessible to a larger audience but the witch doctor rarely wants to share his tricks. Wiki is especially subject to experts writing in the code of their trade with no thought to a larger audience. I could not find these these simpler equations on the Internet. They are shown in the textbooks I reference. The right hand side of the equation is Maxwell's displacement current. The amps in the loop equals the flux of amps through the area of the loop.

Memristors
dq = -I*dt, I is amps. dq, the charge is a current source.
d B = V*dt, V is volts. d B, the magnetic flux, through an area, is a voltage source.

dV = R*dI, R is a resistor and a traditional electrical element.
dq = C*dV, C is a capacitor and a traditional electrical element.
d B = L*dI, L is a inductor and a traditional electrical element.
We have R, C, and L defined by unique groupings of dV, dq, dI and d B. For symmetry we needed a new electrical element M with an additional dq and d B.
d B = M*dq, This was Leon Chua's 1971 IEEE contribution which he called the Memristor, memory resistor. M is memristance. This was spectacularly confirmed with far reaching futuristic electronics in the May 1, 2008 issue of Nature by a team from HP. A reversable amount of current which has flowed through the device programs its resistance. The resistance remains after the power is removed.
(d(B*pi*r2)/dt) /(c*u0) = amps, Faraday's law/(c*u0).
d( B)/dt /(c*u0) = dq/dt, using the flux symbol and dq/dt for the amps.
d( B) = (c*u0)*dq, compare this with the following:
d( B) = M*dq, This is the memristor. M = c*u0 = the memristance.
d( B)/dq = V*dt/(-I*dt) = V/I = M = c*u0,

V/I = 2*pi*r*E/(2*pi*r*B/u0) = E*uo/B = B*c*u0/B = c*u0. The voltage from Faraday's law divided by the amperage from Ampere's law equals the impedance of space which is constant for all wavelengths of electromagnetic radiation. This V/I is a ratio seen in electromagnetic waves not a characteristics of space as the name, impedance of space, would indicate.

Ampere's law = Faraday's law/(c*u0), amps = amps.
B*c = E, in electromagnetic waves.
2*pi*r*B/u0 = 2*pi*r*E/(c*u0), multiply both sides by 2*pi*r/(c*u0). This is, Ampere's law = Faraday's law/(c*u0).
2*pi*r*B/u0 = 2*pi*r*B*c/(c*u0), E = B*c in electromagnetic waves.
2*pi*r*B/u0 = 2*pi*r*B/u0, They are equal. The amps from both laws must be equal for one law to transform into the other. Figure 2Ba: Flux tubes showing their negative excursions as another color. Figure 2D, Flux tubes as sequential machines: These are top and side views of figures 2 and 3B. These are drawn to scale according to the wavelength and fatness ratio which are invarient features of all wavelengths. They all have the same shape. The wavelength here is 314 pixels and the maximum diameter of the flux tubes is 25 pixels. The flux tubes here are pointy jelly beans strung together like sausages. Each wave has a E and B flux tube. The electric field E is a +red and -pink sine wave flux tube. E exerts a torque around B. The magnetic field B is a +green and -cyan cosine wave flux tube. B exerts a torque around E. The cross sectional radius and ring of circumference of the flux tubes is proportional to sine2 or cosine2 as is their energy.

The energy is located along the rotating circular rings of circumference of the flux tubes. E and B rotate in opposite directions. Where the E and B flux tubes touch there is a rolling contact and transfer of energy and circumference as the flux tubes change size. This is a three dimensional view of an action which occurs over time on the two dimensional surface of the expanding spherical wavefront. The action and flux tubes are created by expanding and shrinking ring pairs on the wavefront. We have an alternating electric field flux tube which spirals along an alternating magnetic field flux tube and an alternating magnetic field flux tube which spirals along an alternating electric field flux tube. They are braided. A line between the centers of the E and B flux tubes traces out the icon of life, a double helix. Life preceeded by light. Nature also shows us this shape in a stream of water or the chop on a lake. There is a circular circulation in a cross section of a wave of water or a wave of light. Here the waves only appear bean like or volume like when seen over time, but they exist only as swirling rings of energy on an expanding spherical wave front, only in the here and now, rain drops making rings on still water. Figurer 2D, Flux tubes as sequential machines. Working left to right for the four steps of one wavelength. The loops which were visible, above and below the line on figure 2Ba, are now hard to see on figure 2D. They are clearly shown in the cross sections of figure 2E as the sines and cosines of the triangles.
• Red in green out, red exerts a torque around green, +E → +B, d(sine)/dt → cosine. Row 1 on figure 2E.
e0*d(E*pi*r2)/dt = 2*pi*r*B/u0 = amps, Ampere's law. Maxwell's red changing poloidal displacement current times the area of the loop equals the green toroidal current times the circumference of the loop.
• Green in -pink out, green exerts a torque around -pink, +B → -E, d(cosine)/dt → -sine. Row 2 on figure 2E.
(d(B*pi*r2)/dt) /(c*u0) = 2*pi*r*-E /(c*u0) = amps, Faraday's law/(c*u0). The changing green poloidal current times the area of the loop equals the toroidal pink current times the circumference of the loop.
• -Pink in -cyan out, -pink exerts a torque around -cyan, -E → -B, d(-sine)/dt → -cosine. Row 3 on figure 2E.
e0*d(-E*pi*r2)/dt = 2*pi*r*-B/u0 = amps, Ampere's law. Maxwell's changing pink poloidal displacement current times the area of the loop equals the toroidal cyan current times the circumference of the loop.
• -Cyan in red out, -cyan exerts a torque around red, -B → +E, d(-cosine)/dt → sine. Row 4 on figure 2E.
(d(-B*pi*r2)/dt) /(c*u0) = 2*pi*r*E /(c*u0) = amps, Faraday's law/(c*u0). The changing poloidal cyan current times the area of the loop equals the toroidal red current times the circumference of the loop. Figure 2E, Click image to animate! The plane of the paper is the wavefront. The rings are on the wavefront. Light is rings of current. These rings are the substance and hold the energy of electromagnetic waves. Flux tubes are the integration of these rings over time. Rings preceed flux tubes. Any cross section through the hollow flux tubes show their origin in the rings. The electric field rings move up and down. E is a +red or -pink ring. Its radius is proportional to the sine2. Verticle movement of E exerts a torque around the B horizontal axis. The magnetic field rings move left and right. B is a +green or -cyan ring. Its radius is proportional to the cosine2. Horizontal movement of B exerts a torque around the E verticle axis.

The energy is located along the rotating circular rings of circumference of the flux tubes. E and B rotate in opposite directions. Where the E an B rings touch there is a rolling contact and transfer of energy and circumference as the rings change size. This is an action which occurs over time on the two dimensional surface of the expanding spherical wavefront. The action and flux tubes are created by expanding and shrinking ring pairs on the wavefront. We have an alternating electric field ring which spirals along an alternating magnetic field ring and an alternating magnetic field ring which spirals along an alternating electric field ring. They are braided. A line between the centers of the E and B rings or the flux tubes traces out the icon of life, a double helix. Life preceeded by light.

We saw on figure 2D top and side views showing sine and cosine waves. This is a movie array showing 24 cross sections through those flux tubes per wavelength. Here we have end views of cross sections of figure 2D. The cross sections through the flux tubes are loops or rings. The sequence is +red→+green→-pink→-cyan→+red. The expanding spherical shell of any electromagnetic wavefront would hawidth="443" height="194" ave polka-dots of this pattern. These rings are the only substance an electromagnetic wave has. They only exist on a wavefront.
We only see the sine and cosine waves or flux tubes of figure 2D through the persistance of vision of an oscilloscope. The circles in the animation are vaguely reminiscent of accreation disks, like those found in binary star systems, where one star streams material onto the other star. Here we expect a current to stream from one ring to the other at their point of rolling contact. • On the left, a capacitor ring is shown in two stages as it shrinks.
• A string of charges making a ring has a uniform charge per unit length. The charge Q along the rings per unit length is uniform but the circumference and area is changing. The ring looses charge as it shrinks. A shrinking or expanding ring with a charge per unit length would show dQ/dt which is amps.
• A charge has an electric field E. The electric field along the ring is uniform but the length and area is changing so it shows dE/dt. This is Ampere's law.
• On the right, a magnetic ring is shown in two stages as it shrinks.
• A string of magnetic beads making a ring has a uniform magnetic charge per unit length. The magnetic field B along the ring per unit length is uniform but the circumference and area is changing so it shows dB/dt. This is Faraday's law.

The central axis of the wave is at the right angle of the triangle. The energy is proportional to the radius or circumference of the rings. The electric field E flux ring +red or -pink moves up or down along the y axis and rotates counter-clockwise. The magnetic field B flux ring +green or -cyan moves left or right along the x axis and rotates clockwise. The verticle or horizontal straight line motions of the rings in and out of the right angle on figure 2E are sine or cosine flux tube waves when seen over time from the perpendicular point of view of figure 2D.

Hypothenuse and wavelength: The rings rotate in opposite directions. The ring to ring transfers occur along the line between their centers, the hypotenuse. The hypotenuse rotates tracing out a double helix over time. When the hypotenuse of the triangle is h = wavelength/(8*pi) the sum of the circumference of the ring pairs is 2*pi*h = wavelength/4. The charge content of the rings rotates at the speed of light as the rings transfer energy to their partner. This is a string or ribbon of charge with a length of 2*pi*h = wavelength/4 in a ring orbit, making four ring to ring transfers per wavelength while traveling a distance of one wavelength at the speed of light. The hypotenuse in this figure is about 7_mm so this could be a life size cross sectional diagram of a 7_mm*8*pi = 176_mm wavelength, 1.7_Ghz electromagnetic wave.

h2= x2 + y2,
Pythagorean theorem. h is the hypotenuse of the right triangle in figure 2E and the distance between the rings and the maximum radius of the flux tubes of figure 2D.
x = cos*h = cos*wavelength/(8*pi), distance from the wave central axis to the center of the x ring.
y = sin*h = sin*wavelength/(8*pi), distance from the wave central axis to the center of the y ring.
h2 = h2*cos2 + h2*sin2, substituted for x2 and y2.
h = h*cos2 + h*sin2, multiplied by 1/h. h, the distance between the rings, at wavelength/(8*pi), is constant.
h*cos2 = wavelength*cos2/(8*pi), the radius of the x ring. In the waveforms of figure 2D, the wavelength is 314 pixels and h the maximum radius is 12.5 pixels, so the waveform is drawn to scale.
h*sin2 = wavelength*sin2/(8*pi), the radius of the y ring.
2*pi*h*cos2 + 2*pi*h*sin2 = 2*pi*h = wavelength/4, the sum of the circumferences of the ring pair.

Ring mass, energy and charge are shared between the rings: The end views of figure 2E are useful. We have rings, masses, fluxes and currents which flow in the x and y directions. A verticle current is a currenty. A horizontal current is a currentx.
mt = hp*c/wavelength / c2 = hp/(wavelength*c). mt = total mass of both rings which equals the energy divided by c2. The energy, mass and charge in each ring are proportional to the sine2 or cosine2 of the wave. hp is Plank's constant.
mt2 = x2 + y2, the Pythagorean theorem substituting mt for the hypotenuse of the triangle.
x = mt*cos,
y = mt*sin,
mt2 = mt2 *cos2 + mt2 *sin2,
mt = mt*cos2 + mt*sin2, multiplied by 1/mt. mt is constant with wavelength. The sum of the two mass equivalents of the energy of the rings is mt. mt is shared between both rings.
mx = mt*cos2, mass of x ring.
my = mt*sin2, mass of y ring.

qw = qx + qy, qw the charge is a constant of nature. See appendix 3. qw is shared between both rings.
qx = qw*cos2
qy = qw*sin2 Figure 2F, Graph of sin2 and cos2: as in the radius, energy or masses of the rings. Red, E ring, sin2. Green, B ring, cos2. Blue, E + B rings, sin2 + cos2. Cyan, frequency reference sine wave. h*cos2 or h*sin2, are the radius of the rings. Their graphs are symmetrical. One ring grows and the other shrinks. The sum of the radii or the sum of the circumference of the two rings are constant. The sum of the energy of the two rings are constant. These are twice the frequency of the electromagnetic wave. The height of the graph indicates that the total energy from the sum of E and B is constant while oscillating between E and B. This is consistent with all the charge, energy or mass of the wave being uniformly distributed over the sum of the circumference of the ring pairs which is 2*pi*h = wavelength/4. Any verticle line shows the division of the radius, circumference, charge, mass or of the energy between the colors so we can write,
B2*wavelength3/u0 = hp*c/wavelength = total energy, See appendix 3. hp is Plank's constant.
B2*wavelength3/u0 *(sine2 + cosine2) = hp*c/wavelength, (sine2 + cosine2) = 1
B2*wavelength3/u0 *sine2 + B2*wavelength3/u0*cosine2 = hp*c/wavelength,
B2*wavelength3/u0 *sine2 + E2/c2*wavelength3/u0*cosine2 = hp*c/wavelength, B2 = E2/c2
B2*wavelength3/u0 *sine2 + E2*(e0*u0)*wavelength3/u0*cosine2 = hp*c/wavelength, 1/c2 = (e0*u0)
B2*wavelength3/u0 *sine2 + E2*e0*wavelength3*cosine2 = hp*c/wavelength = total energy, Figure 2G, Graph of the rate of change of sin2 or cos2: as in the rate of change of the radius, circumference, charge, energy or mass of the rings. Red, E ring, (2*sin*cos). Green, B ring, -(2*sin*cos). Cyan, frequency reference sine wave. The rates of change are maximum when the rings have the same size. The rates of change are minimum when the rings have their maximum or minimum radius. In figure 2E, we expect a current to stream from one ring to the other at their point of contact. In figure 2G, above the line current flows in and below the line current flows out. The current flowing out of one ring equals the current flowing into the other ring just like the flow through a junction in hydraulic calculations. A shrinking or expanding ring with a charge per unit length would show a dq/dt = amps ring to ring current.

Rate of change of the circumference of the rings, d(2*pi*r*frequency)/dt: The rate of change of the length of the circumference, d(2*pi*r)/dt, has units of velocity, meters/second.
d(2*pi*r)/dt = d(2*pi*h*sin2)/dt = 2*pi*wavelength/(8*pi)*d(sin2)/dt
d(2*pi*r)/dt *frequency = wavelength/4*d(sin2)/dt*c/wavelength = c/4*d(sin2)/dt = c*sin*cos/2, This rate of change may be plus or minus for a forward d(sin2)/dt or reverse d(cos2)/dt change.

Current in the rings, dq/dt:
Current is the rate of change of the (amps*seconds) charge.
qw = Ce/(2*alpha).5= (hp/(c*u0)).5 = amps*seconds. qw is the total charge. It is spread over the circumference of the rings.
qw/(total length of loops) = qw*4/wavelength, This is the charge per unit length. The sum of the circumference of both loops is the wavelength/4.
dq/dt = qw*4/wavelength *d(2*pi*r*frequency)/dt = qw*4/wavelength *c/2*sin*cos =
dq/dt = qw*c/wavelength*2*sin*cos = qw*frequency*2*sin*cos = amps,

dq/dt = qw*frequency = amps. This is the positive amps, qw*frequency*d(sin2)/dt or the negative amps qw*frequency*d(cos2)/dt, which flow from ring to ring.
This is in the form, charge/meter * meter/second = charge/second = amps*seconds/seconds = amps. How could we know anything without units?

E in the rings, dE/dt:
Anything that has a charge has an electric field. The electric field may point charge to charge, or be generated in a loop like Faraday's law. We will see a loop as a bipolar unit, like a long bar magnet, length of spherical magnets or magnetic beads, whose oppositely charged ends have looped around and stuck together thereby loosing its bipolar character. The charge of the wave is quite small and is spread over the length of the rings. The static electric field due to this small charge is also very small. A shrinking or expanding ring with a charge per unit length would have a dE/dt. The dynamic rate of change of the electric field, which is a product of multiplication of the small charge by the rate of change of the circumference times the frequency, can be very large.
We postulate a further bipolar electric field for the flux units where opposite polarity may hold the flux units into rings. This constitutes a tensile strength associated with the electrical flux, an electrical pinch force. When the units in the rings from figure 4 are held together by this bipolar electric field the electric field is confined within the ring but the perpendicular bipolar magnetic field is exposed.
Etotal/(total length of loops) = Etotal*4/wavelength, This is the E charge per unit length. Etotal is the total E spread over the circumference of the rings. The sum of the circumference of both loops is the wavelength/4.
dE/dt = Etotal*4/wavelength *d(2*pi*r*frequency)/dt = Etotal*4/wavelength *c/2*sin*cos =
This is in the form, E/meter * meter/second = E/second = dE/dt
dE/dt = Etotal*c/wavelength*2*sin*cos = Etotal*frequency*2*sin*cos
dE/dt = 4*pi*E*frequency. This is Ampere's law if Etotal*2*sin*cos = 4*pi*E. 2*sin*cos = d(sin2)/dt.

B in the rings, dB/dt:
A string of magnetic beads has a magnetic charge per unit length. A shrinking or expanding ring with a magnetic charge per unit length would show a dB/dt.

Magnetic pinch force:
A ring of bipolar magnetic beads shows tensile strength. Is this the magnetic pinch force? We postulate a static bipolar magnetic field for the flux units where opposite polarity may hold the flux units into rings in this same way. When the units in the rings from figures 4 are held together by this bipolar magnetic field the magnetic field is confined within the ring but the perpendicular bipolar electric field is exposed.

Btotal/(total length of loops) = Btotal*4/wavelength, This is the B charge per unit length. Btotal is the total B spread over the circumference of the rings. The sum of the circumference of both loops is the wavelength/4.
dB/dt = Btotal*4/wavelength *d(2*pi*r)/dt = Btotal*4/wavelength *c/2*sin*cos =
This is in the form, B/meter * meter/second = B/second = dB/dt.
dB/dt = Btotal*c/wavelength*2*sin*cos = Btotal*frequency*2*sin*cos
dB/dt = 4*pi*B*frequency. This is Faraday's law if Btotal*2*sin*cos = 4*pi*B. -2*sin*cos= d(cos2)/dt.

New currents: The rate of change of the charge on each ring as the rings change size constitutes a current which flows from ring to ring across the plane of the wavefront. The extremes of the waves, on figure 2G, are the maximum currents flowing ring to ring which are,
dq/dt = qw*frequency = amps.
dB/dt = 4*pi*B*frequency.
dE/dt = 4*pi*E*frequency. We haven't assigned names to the dB/dt and dE/dt currents.

The link between rings on the wave front and their perpendicular waves: We have demonstrated toroidal currents which flow ring to ring on the wavefront. We must also describe these same currents flowing forward at the speed of light with the wavefront through the flux tubes according to Faraday's and Ampere's law. When Ampere's law was written to show Maxwell's displacement current we noticed, "The toroidal amps in the loop equals the poloidal flux of amps through the area of the loop." This is the link between the toroidal current flowing ring to ring on the wavefront and the poloidal current flowing through the area perpendicular to the wavefront. Ampere and Faraday are about perpendicular relationships. The toroidal current flowing ring to ring on the wavefront equals the perpendicular polodial current flowing through the flux tubes.
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Is energy stored in the area or the circumference of the flux tubes? At the cosmic scale, objects are mostly volume and little surface. At the smallest scale, objects are mostly surface and little volume. Volume/surface of a sphere = radius/3. For the cosmos, radius/3 = 4.73E25_m. For red light, wavelength/3 = 211E-9_m. At the smallest scale, objects are mostly circumference and little area. Area/circumference of a circle = radius/2. For red light, wavelength/2 = 316E-9_m. The circumference is 3 million times bigger than the area. We would expect circumference to be much more important. The circumference of the ring pair does carry the charge. We previously noticed that when Ampere's law was written to show Maxwell's displacement current, "The toroidal amps in the loops equals the poloidal flux of amps through the area of the loop." Both area and circumference are important.

Rate of change of the area of the rings, d(pi*r2)/dt:
pi*r2 = pi*(h*sin2)2 = pi*h2*sin4 = wavelength2*sin4/(64*pi): h = wavelength/(8*pi)
pi*r2 = pi*(h*cos2)2 = pi*h2*cos4 = wavelength2*cos4/(64*pi): h = wavelength/(8*pi)
d(pi*r2)/dt = wavelength2*cos*sin3/(16*pi), The rate of change of the area of the rings.
d(pi*r2)/dt = wavelength2*sin*cos3/(16*pi), Does this go anywhere? Figure 2H, Graph of the area of the rings: Red, E ring, sin4. Green, B ring, cos4. Blue, E + B rings, sin4 + cos4. Cyan, frequency reference sine wave. The area of the rings are pi*r2 or pi*h2*sin4 or pi*h2*cos4. The graph of sin4 and cos4 are not sine waves as is their sum. The sum of the area of the rings is the elevated blue sine wave which oscillates around a value at four times the frequency of the wave. One might say they shimmer. Is this a residual field? Figure 2J, Graph of rate of change of sin4 and cos4: as in the rate of change of the area of the rings. Red, E ring, 4*cos*sin3. Green, B ring, 4*sin*cos3. Cyan, frequency reference sine wave.

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Figure 4: Click to animate! What is flux? The cross sections through the flux tubes, in the movie, are rings. Only two rings can exist at a time but we have shown all four rings so you can see their rotations. These rings evolved from earlier work on spirals. See the spiral animated gif. The rings, at this scale, are charged colored ribbons. They unroll from one ring and roll onto the next ring. They transform where the ribbon changes colors, E becomes B or B becomes E. +red→+green→-pink→-cyan→+red. The color changes where the rings touch. Visualize a uniform charge distributed over the variable circumference of both rings. The rate of change of the charge is a current transfered ring to ring.

The hypotenuse in this figure is about 37_mm so this could be a life size cross sectional diagram of a 37_mm*8*pi = 930_mm wavelength, 322_Mhz electromagnetic wave. This detailed mechanistic view of electromagnetic waves makes falsifiable predictions. Standing waves of figure 2D have a fixed spacing of E and B fields. The E fields may be measured and located and the B fields inferred. Properly spaced B fields of a certain strength would apply a predictable polarizing torsion.

The electric field flux tube, in cross section, rotates counter-clockwise. The magnetic field flux tube, in cross section, rotates clockwise. This suggest a simple root for their differences. Inertia can be understood as the acceleration dependent gravitational influence of the background cosmos. We might understand the differences between the E an B fields in terms of the fields and rotation of the background cosmos. Figure 4A: The rings and ribbons are shown face on, as in figure 4 and figure 2E. These ring transitions are shown here at a larger scale and are unrolled flat. The face and bottom have opposite charges. The sides, where the rotation arrows are shown, have little or no E or B charge. Figure 4B: The rings and ribbons are shown edge on. This shows the hypothetical units of charge and flux. The ribbons and rings are made from a long string of these units of flux. They each have a bipolar magnetic B field and a perpendicular bipolar electrical E field. Each unit is a flat square having a fractional a*s charge per unit and four E or B charged edges: -E,-B,+E,+B. These are series and parallel magnets and charges. We have series magnets which stick together while sporting parallel charges. We have series charges which stick together while sporting parallel magnets. They interact like square magnets, opposites attract, likes repell, B and E ignore each other. The ring shaped stack of units which is unrolling radiates units. The units transform when each unit rotates and restacks. E becomes B or B becomes E by showing a different pair of edges. The edge pairs, which stick together, B and -B or E and -E, merge into a ring. The strong bipolar fields of the merged units are concealed within the rings. What remains is a ring, which still shows a charged top and bottom from the other perpendicular bipolar field, B and -B or E and -E. Stacks of the units have opposite charges on their ends. The ends attract each other to loop around to make rings.

Pinch and repulsion:
A row of parallel magnets, with their bi-poles pointing in the same direction, repell each other. If the parallel magnets are each rotated ninety degrees, they are in series, their poles now attract each other. They form rows or rings of magnets with a tensile strength. Call this opposite pair magnetic pinch and magnetic repulsion.

A row of parallel bipolar charges, with their bi-poles pointing in the same direction, repell each other. If the parallel bipolar charges are each rotated ninety degrees, they are in series, their poles now attract each other. They form rows or rings of bipolar charges with a tensile strength. Call this opposite pair electrostatic pinch and electrostatic repulsion.

Units may have both bipolar magnetism and bipolar charge arranged in a cross or square which are assembled into rows or chains which assemble into rings. The units may rotate in ninety degree steps and make rows and rings as follows: We have magnetic force holding the row together, which is concealed, except for the ends which are also concealed when this loops into a ring, while showing perpendicular charge plus up. We have electrostatic force holding the row together, which is concealed, except for the ends which are also concealed when this loops into a ring, while showing perpendicular magnetism plus up. We have magnetic pinch holding the row together, which is concealed, except for the ends which are also concealed when this loops into a ring, while showing perpendicular charge plus down. We have electrostatic force holding the row together, which is concealed, except for the ends which are also concealed when this loops into a ring, while showing perpendicular magnetism plus down. We have magnetic force holding the row together, which is concealed, except for the ends which are also concealed when this loops into a ring, while showing perpendicular charge plus up.
Can you see the flux tubes of figure 2D and 2E and the rings of figure 4 in terms of series and parallel bipolar charges?

Rings of magnetic beads or spherical magnets: have a lot of tensile strength. They are hard to pull apart. This must be related to the magnetic pinch force. Rings hide the bipolar glue of the units which holds them together in rings. Their hidden flux is confined to the ring. Rings of very strong spherical magnets have a very weak external magnetic field. See the Beaty video. Magnets have other interesting structural assembly properties. Interesting sources are K&J and neocube. Warning! Magnets can be addictive. One might be subject to spousal abuse for spending too much money on too many magnets. Rings and iron filings:
The ribbons share a lot with the field pattern of iron filings on a glass over a bar magnet. The field lines are made from tiny magnets made of the magnetized iron filings which are held together by the tensile strength of the magnetic field. The field lines of iron filings are held apart by the repulsion between the lines of tiny magnets. Experiments with spherical magnets show similar behavior. The field pattern of grass seeds in an insulating liquid or bits of thread in oil, between oppositly charged rods, shows the same pattern as the bar magnet and iron filings. Halliday and Resnick or Tipler have great pictures. Also see the pictures of sources and sinks. How could the particles stick together without opposite charge on opposite ends? What geometry allows an electron to show only its negative charge? The rings of figure 5 are a good guess. Figure 5: This is loops of light as particles. The outside of the rings are, left to right, +E,-E,+B or -B. The inside of the rings is hidden. There are four possible arrangements. These particles would attract or repell each other. The first two are oppositely charged particles with a bipoler magnetic field like electrons and positrons. The first two would attract each other, stack and stick together. The second pair are magnetic monopoles which have a bipoler electrical field. The second two would attract each other, stack and stick together. The pair of monopoles stack becoming bipoles. These are possible aspects of light which is deflected. We know that light can be deflected from numerous cases of gravitational lensing. These rings will illustrate our point but light can be deflected without being deflected into a ring. On the left, the green and cyan wave is deflected but the red and pink wave is deflected and also has a larger outside diameter than its inside diameter. The energy density and the rate of change of the wave on the inside diameter is greater than on the outside diameter. These are magnetic or electric gradients. This imbalance has residual effects, one of which may be gravity. See Electrostatic gravity. This search for the origin of gravity is also articulated by A.K.T. Assis. His books and online papers are recommended.

Antenna analogy: We equate the amp form of Ampere's law, on the left, with the amp form of Faraday's law.
2*pi*r*B/u0 = 2*pi*r*E/resistance = amps
resistance = E*u0/B, divided by 2*pi*r
resistance = B*c*u0/B, E = B*c in electromagnetic waves.
resistance = c*u0 = 376.73 ohms, cancelled B

In Faraday's law a current is produced in a loop or coil: when the electric field loops around, or when the magnetic field varies, or the area of the loop varies. Without wires in an electromagnetic wave we see the impedance of free space, c*u0 = (u0/e0).5 = 376.73 ohms. 300 ohm antenna lead wires are matched to this impedance so that the antenna may transfer maximum energy.

Does space have resistance? Space is empty. If space had any resistance, electromagnetic waves would be damped quickly, dissipate their energy as I2R heat, instead of traveling for billions of light years. The impedance of space, is more properly described as shorthand for the ratio, V/I = Faraday's voltage divided by Ampere's current in electromagnetic waves = 376.73 ohms. Electromagnetic waves have this V/I ratio so antennas should be most efficient when matched to this V/I ratio. This ratio is called impedance because it has units of V/I or ohms.

Antenna theory says accelerated charges radiate. Changes in direction are regarded as accelerations so something in a circular orbit is accelerated. Accelerated charges are changing amps which are produced by d(B)/dt. The flux of B through the area or rate of change of the circumference of the loop radiates the -E seen in the loop. Arcing, which is the radiation or emission of currents, occurs at a lower potential from pointed objects, those with a smaller radius of curvature. The radius of curvature of the flux or of the loop is the wavelength/(2*pi) which can be very small so the radiation or emissions can be almost instantaneous. The transition is proportional to the wavelength. See appendix 1.

__________________________________________________________________ Figure 6: This is a stack of cross sections through the electromagnetic wave as it covers one wavelength. The lines in the middle are a reference between the circle which is expanding and the circle which is shrinking, the hypotenuse of the right triangle of figure 2E. This is a summation of the movement in the movie and the movie array. Science as art.

Euler's equations

Euler's equations and electromagnetic wave dynamics:
Euler's equations are used to express three dimensional rotational motions; roll, pitch and yaw in aircraft or spacecraft and precession or nutation in gyroscopes and rotating bodies. We use them to understand the flux and looping around of Faraday's and Ampere's laws in electromagnetic waves and light. Top view, top: Red exerts a torque around the green axis, which is into the page, in the top view.
Side view, bottom: Green exerts a torque around the red axis, which is into the page, in the side view. These waveforms are not the right shape. They do not go to zero at a change in direction. They are fun to make out of colored wires, see Appendix 5. The red and green are correctly, 90 degrees out of phase. They demonstate the torque around a perpendicular axis. A torque around two perpendicular axises produces a torque perpendicular to both axises in a gyroscope. Here the axises of the torque is moving with the waves. How does this work? This is gyroscopic precession and movement in the direction of travel of the wave. This is a very odd gyroscope indeed.

Te
Tb Ta
The subscript is for the axis. a is the roll or axial axis which transfers the angular momentum or spin of the light. b is the pitch or magnetic axis which is perpendicular to the axial axis. e is the yaw or electrical axis which is perpendicular to both the axial and magnetic axis.

E exerts a torque around the b axis. B exerts a torque around the e axis. E+B exerts a torque around the a axis. The transfer of torque as angular momentum or spin is along the a axis.

Ta = Ia*dwa/dt +(Ib-Ie)*we*wb
Tb = Ib*dwb/dt +(Ie-Ia)*wa*we
Te = Ie*dwe/dt +(Ia-Ib)*wb*wa

w is angular velocity.
dw/dt is the the angular acceleration, the rate of change of the angular velocity.
I is the moment of inertia, I = mass*radius2, for the hoop or ring which we see along the spherical wavefront in figure 2E. The mass is only that which is calculated from the energy. The total mass mt, is proportional to energy and therefore proportional to E2 or B2 of the energy density and sin2 or cos2 of our waves.
T is the torque or moment. torque = moment of inertia * angular acceleration.
h is the hypothenuse, of the right triangle, on figure 2E.

The moment of inertia: is calculated using the parallel axis theorem,
I = Icm + m*D2, Icm is the center of mass moment of inertia of the ring Icm = m*r2. m is the mass.r is the ring radius. D is the distance Icm moves from the central axis of the wave in the oscillation of the E and B rings. See figure 2E.
I = Icm + m*D2, I = m*r2 + m*D2, I = m*(r2 + D2),
Ib = mt*cos2 * (h2*cos4 + h2*cos2) = mt*h2*(cos6+cos4), m = mt*cos2, r = h*cos2, r2 = h2*cos4, D = h*cos, D2 = h2*cos2
Ie =mt*sin2 * (h2*sin4 + h2*sin2) = mt*h2*(sin6+sin4), m = mt*sin2, r = h*sin2, r2 = h2*sin4, D = h*sin, D2 = h2*sin2
The moment of inertia Ia is due to Ib and Ie.
Ia = Ib + Ie = mt*h2*(cos6+cos4) + mt*h2*(sin6+sin4) = mt*h2*(cos6+cos4+sin6+sin4)

If we have waves traveling unimpeded: wa = wb = we = w, since the angular velocity w is constant, the rate of change of the angular velocity, the angular acceleration, d(w)/dt = 0, and each equation is simplified.
Ta = (Ib-Ie)*we*wb
Ta = (mt*h2*(cos6+cos4) - mt*h2*(sin6+sin4))*w2
Ta = mt*h2*w2*(cos6+cos4 - sin6-sin4)

Tb = (Ie-Ia)*wa*we
Tb = (mt*h2*(sin6+sin4) - mt*h2*(cos6+cos4+sin6+sin4))*w2
Tb = mt*h2*w2*(-cos6-cos4)

Te = (Ia-Ib)*wb*wa
Te = (mt*h2*(cos6+cos4 + sin6+sin4) - mt*h2*(cos6+cos4)*w2
Te = mt*h2*w2*(sin6+sin4)

Ta + Tb + Te = mt*h2*w2*((cos6+cos4-sin6-sin4) +(-cos6-cos4) +(sin6+sin4)) = 0,
The sum of the torques is zero. Figure 7: Unimpeded torques of the light wave. Ta, blue, (cos6+cos4-sin6-sin4). Tb, green, (-cos6-cos4). Te, red, (sin6+sin4). Cyan, reference sine wave. The sum of the torques is zero while the waves travel unimpeded through space.

If the waves are impeded: or impeded while being detected the d(w)/dt terms are no longer zero.
Ta = Ia*dwa/dt + (Ib-Ie)*we*wb
Ta = mt*h2*(cos6+cos4 + sin6+sin4)*dwa/dt + mt*h2*(cos6+cos4 - sin6-sin4)*we*wb
Ta = mt*h2 *[(cos6+cos4 + sin6+sin4)*dwa/dt + ((cos6+cos4 - sin6-sin4)*we*wb)] = mass torque or spin.

Tb = Ib*dwb/dt + (Ie-Ia)*wa*we
Tb = mt*h2*(cos6+cos4)*dwb/dt + mt*h2*(-cos6-cos4)*wa*we
Tb = mt*h2*(cos6+cos4) *[dwb/dt + (-1*wa*we)] = magnetic torque

Te = Ie*dwe/dt + (Ia-Ib)*wb*wa
Te = mt*h2*(sin6+sin4)*dwe/dt + mt*h2*(sin6+sin4)*wb*wa
Te = mt*h2*(sin6+sin4) *[dwe/dt + (wb*wa)] = electrical torque

The waves change their energy content or dump their energy as photons through angular momentum = spin, magnetic torque or electrical torque. The later two do their work with Faraday's and Ampere's laws. This is the primary mechanism for wave-wave or wave-particle interactions.

Appendix

Appendix 1: Transition math:
frequency*wavelength = c
wavelength = 4*transition, so
frequency*transition = c/4, At higher frequencies the transition is shorter.
energy = hp*frequency, hp is Plank's constant.
energy*transition = c*hp/4, At higher energies the transition is shorter.

Energy or frequency changes result in tensile or compressive forces. The circumference and rate of change of the circumference, of the rings of figure 2E, changes. Electromagnetic waves can be viewed as a coiled spring. When the wavelength increases the distance between the coils increases. This is a tensile force on the medium. When the wavelength decreases the distance between the coils decreases. This is a compressive force on the medium. There can be huge forces, at high currents anywhere the wavelength or frequency varies. Exploding wires which look like fragmented spaghetti and compression damage in rail guns have been noted. See Graneau and Graneau in, "Newton versus Einstein" for these and other details of the ongoing conflict between theory and experiment.

Appendix 2:
E = force/q, forceE = q*E
B = force/(q*v), forceB = q*v*B
forceE = forceB
q*E = q*v*B:
J. J. Thomson determined the mass to charge ratio of the electron using this equation. This is similar to the Lorentz force = q*E + q*v*B if the forces are equal. We only use it with the velocity v equal to c, the charge moves at the speed of light and the two forces are equal.
The electrostatic force of repulsion by the charge equals the magnetic pinch force of attraction on the charge. We see it in the ring electron and the electromagnetic wave.
E = c*B, cancelled q, units are volts per meter or kg*m/(A*s3)
E/B = c
E2/B2 = c2,
square
E2/B2 = 1/(u0*e0), c2 = 1/(u0*e0)
E2*e0 = B2/u0, kg/(m*s2). The B and E energy densities or pressures are equal. This is the magnetic pinch pressure equals the electrostatic pressure of repulsion. This magnetic pinch pressure restrains the charge to the thin flux tube ring of the electron like a hose restrains water.

Appendix 3: Derivation of the wave parameters:
There are four variables with values that we seek; B, E, q and amps.
2*pi*r*E = volts = energy/q, Faraday's law
E*q = energy/2*pi*r, force, energy/wavelength, q = charge
E*q = hp*c/(wavelength*2*pi*r), force, energy = hp*c/wavelength, hp is Plank's constant.
B*q = hp/(wavelength*2*pi*r), kg/s, B*c*q = E*q

2*pi*r*B = amps*u0, Ampere's law, B = force/(amps*meters)
B = amps*u0/2*pi*r or B = amps*u0/wavelength. kg/(a*s2)

B*q / B = hp/(wavelength*2*pi*r) * 2*pi*r/(amps*u0)
q1 = hp/(wavelength*amps*u0), u0 = 4*pi/10000000_kg*m/(a2*s2)

q*amps = hp/(wavelength*u0),
energy = hp*c/wavelength = c*u0*q*amps, c, uo and q are constants so energy or mass is proportional to amps.

2*pi*r*B = force/amps, Ampere's law
2*pi*r*B = B*c*q/amps, substituted for force
q2 = amps*2*pi*r/c or q = amps*wavelength/c

q2 = q1,
amps*2*pi*r/c = hp/(wavelength*amps*u0),
amps2*wavelength*2*pi*r = c*hp/u0,
amps*(wavelength*2*pi*r).5 = (c*hp/u0).5, 2*pi*r = wavelength.

(c*hp/u0).5 = 3.976E-10_A*m, amps*wavelength is a constant.

amps = (c*hp/u0).5/(wavelength*2*pi*r).5 or amps = (c*hp/u0).5/wavelength = 3.976E-10_A*m/wavelength. This is the maximum amps. There is a variable current which flows between the rings of figure 2E as shown on figure 2G.

q2 = amps * 2*pi*r/c,
q = (c*hp/u0).5/(wavelength*2*pi*r).5 * 2*pi*r/c, substituted for amps
q = (c*hp/u0).5/wavelength.5 * (2*pi*r).5/c, =
q = (hp/(c*u0)).5 * (2*pi*r/wavelength).5, =
qw = (hp/(c*u0)).5 = (hp*c*e0).5 = 1.326E-18_A*s, charge. We see that qw, the charge of the electromagnetic wave, is a constant that does not vary with wavelength if 2*pi*r = wavelength.
qw2 = hp/(c*u0)

(c/(u0*hp)).5 = 6.000359E23_a*s/(kg*m), curiously close to Avogadro's number

amps = qw*frequency
amps = qw*c/wavelength
hp = qw2*c*u0 = qw2/(c*e0), qw2 = hp/(c*u0)
hp = qw2/(c*e0), hp is Plank's constant. qw is the charge of electromagnetic waves.
hp = Ce2/(2*c*e0*alpha), Ce is the charge of the electron. Alpha is the fine structure constant.
Ce2/(2*c*e0*alpha) = qw2/(c*e0), hp = hp.
Ce2 = qw2*2*alpha,
qw = Ce/(2*alpha).5, This qw is 8.277 times the charge of the electron. Why is this so? This is the total charge qw which is shared between the E and B rings.
qw = qx + qy = qw*cos2 + qw*sin2, This is the division of the charge in the x y plane between the two rings.

B = amps*u0/2*pi*r, kg/(a*s2). Ampere's law.
B = (c*hp/u0).5/(wavelength*2*pi*r).5 *uo/(2*pi*r), substituted for amps

(c*hp*u0).5 = hp/qw = 4.996E-16_kg*m2/(A*s2), energy/amps is constant.

B = (c*hp*u0).5/(wavelength*2*pi*r).5/(2*pi*r) or B = (c*hp*u0).5/wavelength2.

B2/u0 = energy density
B2/u0 = ((c*hp*u0).5/wavelength2)2/u0
B2/u0 = (c*hp*u0)/wavelength4/u0
B2/u0 = c*hp/wavelength4

B2*wavelength3/u0 = c*hp/wavelength, energy. This is energy density times wavelength cube.

E = c*B = c*(c*hp*u0).5/wavelength2.
E2*e0 = energy density
E2*e0 = (c*(c*hp*u0).5/wavelength2)2*e0
E2*e0 = c3*hp*u0*e0/wavelength4,
E2*e0 = c*hp/wavelength4, u0*e0 =1/c2

E2*wavelength3*e0 = c*hp/wavelength, energy. This is energy density times wavelength cube.

Powers of 1/wavelength:
energy = hp*c/wavelength, one over wavelength
amps = qw*frequency = qw*c/wavelength = (hp/(c*u0)).5 *c/wavelength, one over wavelength
B = (c*hp*u0).5/wavelength2, One over wavelength2
E = B*c = c*(c*hp*u0).5/wavelength2, One over wavelength2
dB/dt = 4*pi*B*frequency = 4*pi*(c*hp*u0).5/wavelength2*c/wavelength = 4*pi*c*(c*hp*u0).5/wavelength3, One over wavelength3
dE/dt = 4*pi*E*frequency = 4*pi*c*(c*hp*u0).5/wavelength2*c/wavelength = 4*pi*c2*(c*hp*u0).5/wavelength3, One over wavelength3
B2/u0 = hp*c/wavelength4, One over wavelength4
E2*e0 = hp*c/wavelength4, One over wavelength4

Red light example:
wavelength = 633E-9_m, for red light
frequency = c/wavelength = 4.736E14_1/s
qw = (hp/(c*u0)).5 = 1.326E-18_A*s, charge
amps = (c*hp/u0).5/wavelength = qw*frequency = qw*c/wavelength = 6.281E-4_A,
B = (c*hp*u0).5/wavelength2 = 1.2469E-3_kg/(A*s2), Teslas
dB/dt = 4*pi*B*frequency = 7.421E12_kg/(A*s3), Teslas/second
E = c*(c*hp*u0).5/wavelength2 = 373815_kg*m/(A*s3), volts/meter
B2/u0 = E2*e0 = c*hp/wavelength4 = 1.237_kg/(m*s2), energy density or pressure
B2/u0*wavelength3 = 3.318E-19_kg*m2/s2, energy

Electron gamma ray example:
me*c2 = hp*c/wavelength = 8.187E-14_kg*m2/s2, energy. me = mass of the electron
wavelength = hp*c/(me*c2) = hp/(me*c)= 2.4263E-12_m, of the gamma ray
frequency = c/wavelength = 1.235E20_1/s
qw = (hp/(c*u0)).5 = 1.326E-18_A*s, charge
amps = (c*hp/u0).5/wavelength = qw*frequency = qw*c/wavelength = 163.865_A,
B = (c*hp*u0).5/wavelength2 = 8.4869E7_kg/(A*s2), Teslas
dB/dt = 4*pi*B*frequency = 1.317E29_kg/(A*s3), Teslas/second
E = c*(c*hp*u0).5/wavelength2 = 2.5443E16_kg*m/(A*s3), volts/meter
B2/u0 = E2*e0 = c*hp/wavelength4 = 5.7319E21_kg/(m*s2), energy density or pressure
B2/u0*wavelength3 = 8.187E-14_kg*m2/s2, energy

Ring electron and electron gamma ray:
A and qw are 1/(2*alpha).5 = 8.277 time bigger in the electron gamma ray than the ring electron. B and E are 2*pi/(2*alpha).5 = 52.009 times bigger in the ring electron than in the gamma ray. Ring electron energy density me*c2/3.853E-41_m3 = 2.12E27_kg/(m*s2) is 3.69E5 times bigger. Ring electron density is 2.364E10_kg/m3. Nuclear density is 42 billion times larger at 10E21_kg/m3.

Appendix 5: Figure 8A: Wires can be woven together to make quite good toy sine and cosine waves. Ribbons, felt or foam strips can also be used. Physical models are the best analogs for reality. I make the individual sine waves out of different colors of 12 or 14 gage solid core copper wires. I bend the wires around two nails held in a vise and move the "s" shape along while bending until the sine wave is complete. I then weave two of the sine waves together which produces a sine and cosine combination. This figure shows the top and side views of the resulting weaved wires. Tactile sensation amplifies visual sensation. Figure 8B: This shows the construction of the figure 8A. I wrote a Liberty Basic program to draw the circles. A Paint program was used to color and clean up. Almost all of the black lines are removed by using Paint several times to fill the background first with black and then with white. The rest of the graphics and movie in this paper were also Basic programs and Paint.

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Tipler's reciprocal results:
"Physics for scientist and engineers", 4th. ed., p. 1000-1001; "Maxwell's modification of Ampere's law shows that a changing electric flux produces a magnetic field whose line integral around a curve is proportional to the rate of change of the electric flux. We thus have the interesting reciprocal result that a changing magnetic field produces an electric field (Faraday's law) and a changing electric field produces a magnetic field (generalized form of Ampere's law)."

Fat text books can be articulate old friends:
Tipler's; "Physics for scientist and engineers" or Halliday and Resnick's; "Physics for students of science and engineering", both offer the comprehensive coverage and detail useful in understanding this field. They are not dumbed down. The internet is like Lake Okeechobee with its shallows miles wide and its occasional pockets of deep water. Is this as shallow as a mud flat or is this a pocket of deep water? Rose Anne says this website is like the mathematician in "A Beautiful Mind" putting his letters in an unused mailbox for pickup by imagined readers.

Storyland: Before there was writing there were stories. Theories are little stories we use to think about and describe reality; scientific, political or otherwise. Apparently we prefer our ideas served on the platter of a story. Something in us wants us to believe a story. Repetition makes the heart grow fonder. These are addictive memes. We are even seduced by a weak story. A story is an audio, visual, sensory experience as required by neuromarketing. Brain scans would show stories activate pleasure centers while facts do not. There is a narcotic effect in the mantra or in constant repetition of stories. The opiate of the masses is endorphine based. This makes us vulnerable to manipulation. Fancy theories, flag wavers, fundamentalist and fanatics all have their stories. The best stories to believe are based on evidence from multiple sources. Some, which are widely accepted, have only hearsay (he said) evidence or anecdotal (more little stories) evidence. Some, which are properly called dogma, are said to be accepted without evidence, as a requirement for membership in a group. Monkey see, monkey do. We are primates if you prefer the story of evolution instead of the story of Noah's ark. Parotting a plethora of preposterous stories, taken unquestioned at face value, papers ones reality with a crazy quilt of pernicious perculating absurdities. There are so many zombies addicted to stories, so many sacred cows, so many mad dogs ready to kill, if your story is different from theirs. Humor them. We seek clarity, (a clear simple story - like the following).

My son gave me a LED flashlight which uses Faraday's law. A magnet moving bi-directionally in a tube, through a coil of wire, provides a reversing current. That current charges a energy storage capacitor through diodes that keep the reversing current flowing in one direction. The capacitor acts like a battery to light the LED.

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