## Generalizing Coulomb’s Law

The forces between two electric charges in arbitrary motion are complicated by velocity, acceleration, and time-delay effects. The forces need not even lie along the line joining the two charges!

Suppose a source charge $q^\prime$ is at position $\vec r^{\,\prime}$ with velocity $\vec v^{\,\prime}$ and acceleration $\vec{a}^{\,\prime}$, and a test charge $q$ is at position $\vec r$ with velocity $\vec v$ and acceleration $\vec{a}$. Let their separation $\vec{\mathcal{R}}= \vec r - \vec r^{\,\prime}$. Since electromagnetic “news” travels at light speed $\color{blue}c\color{black} = 1$ (in natural units), define the earlier retarded time $s < t$  implicitly by $t - s = \mathcal{R}[s] / \color{blue}c\color{black} = \mathcal{R}_{\color{red}s}$.

Liénard-Wiechert electric and magnetic potentials at test charge $q$ due to source charge $q^\prime$ are

$$\varphi_{\color{red}t} =\frac{q^\prime_e}{4\pi \mathcal{R}_{\color{red}s} }=\frac{q^\prime}{4\pi \mathcal{R}_{\color{red}s} } \left( \frac{1}{1 - \hat{\mathcal{R} }_{\color{red}s} \cdot \vec v^{\,\prime}_{\color{red}s}} \right), \hspace{1cm} \vec A_{\color{red}t}= \varphi^\prime_{\color{red}s} \vec v^{\,\prime}_{\color{red}s},$$

where the source charge $q^\prime$ is “smeared” to an effective charge $q^\prime_e$ by its motion, and the subscripts indicate evaluation times. If the source charge velocity $\vec v^{\,\prime}_{\color{red}s} = 0$, then the potentials simplify to the electrostatics limit. Differentiate the potentials to find Maxwell’s electric and magnetic fields

$$\vec{\mathcal{E}} = - \vec \nabla \varphi - \partial_t \vec A \hspace{1cm} \vec{\mathcal{B}} = \vec \nabla \times \vec A,$$

and the Lorentz force law

$$\vec F = q \left( \vec{\mathcal E} + \vec v \times \vec{\mathcal B}\right)$$

generalizes Coulomb’s law to

$$\vec F_{\color{red}t} = q \frac{ q^\prime } { 4 \pi \mathcal{R}^2_{\color{red}s}} \left( \frac{ 1 } {1 - \hat{\mathcal{R}}_{\color{red}s} \cdot \vec{v}^{\,\prime}_{\color{red}s} }\right)^3 \bigg( \frac{}{} (1-v^{\prime 2}_{\color{red}s})(\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) + \vec{\mathcal{R}}_{\color{red}s} \times \left((\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) \times \vec{a}^{\,\prime}_{\color{red}s} \right)$$ $$\hspace{3.4cm} + \,\vec v_{\color{red}t} \times \left( \hat{\mathcal{R}}_{\color{red}s} \times \left( (1-v^{\prime 2}_{\color{red}s})(\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) + \vec{\mathcal{R}}_{\color{red}s} \times \left((\hat{\mathcal{R}}_{\color{red}s}-\vec{v}^{\,\prime}_{\color{red}s}) \times \vec{a}^{\,\prime}_{\color{red}s} \right) \right) \right) \, \bigg).$$

If the test charge velocity $\vec v = \vec 0$, then the magnetic terms vanish; if the source charge acceleration $\vec{a}^{\,\prime} = \vec 0$, then the radiation terms vanish; if, in addition, the source charge velocity $\vec v\,^\prime = \vec 0$, then the generalized Coulomb’s law reduces to the familiar electrostatic limit. Finally, Newton’s second law with Einstein’s momentum

$$\vec F = \frac{d\vec p}{dt} = \frac{d}{dt} \frac{ m \vec v }{\sqrt{1 - v^2}} = \frac{ m \vec a}{\sqrt{1 - v^2}} + \frac{ m \vec v}{(1 - v^2)^{3/2}} \vec v \cdot \vec a$$

generates the motion equations using 3-vectors and lab time (instead of 4-vectors and proper time), succinctly summarizing all of electromagnetism.

Using Mathematica, I eliminated present time $t$ from the motion equations with the substitution $t = s + \mathcal{R}_{\color{red}s}$ and numerically integrated them with respect to the retarded time $s$ for various initial conditions, as in the figure below. Positive and negative charge pairs tugged by non-zero net forces spiral into one another as they radiate away energy and momentum, graphically demonstrating the instability of classical atoms.

Mathematica simulations of interacting charged particle including magnetic, radiation, and time-delay effects. Blue dots mark present positions, and red dots mark retarded positions. Arrows represent force pairs, which are typically not “equal but opposite”.

## Wooster Physics Alumni at Kent Displays

Three Wooster physics alumni who all work at Kent Displays, Inc. returned to campus last Thursday to share some info about the physics of liquid crystals as well as some of their personal journeys to Wooster and beyond.

See more information over at the NEWS page on the Wooster Physics website!

## Triple EVA

Since the mid 1960s, all space walks or extra-vehicular activities (EVAs) have involved just one or two astronauts — except once. In May 1992, on the STS-49 mission, the crew of the space shuttle Endeavour was attempting to rescue a stranded communications satellite, Intelsat 603. After repeated two-person EVAs failed to capture the satellite, NASA and the crew improvised the world’s first (and so far only) 3-person EVA. Anchored in the shuttle’s cargo bay, astronauts Richard Hieb, Thomas Akers, and Pierre Thuot simultaneously grabbed the large spinning Intelsat by hand. The shuttle crew later attached Intelsat 603 to a live rocket motor and released it, enabling the satellite to reach its desired geostationary orbit, where it successfully operated for 23 years.

During the world’s first (and so far only) 3-person EVA, space shuttle astronauts salvage a stranded communications satellite. (1992 May 13, NASA.)

## 5-Color Theorem

On 1852 October 23, Francis Guthrie noticed that he needed only 4 colors to color the counties of England so no two bordering counties shared the same color. This works for any map, but only in 1976, and with the aid of a computer, did Kenneth Appel and Wolfgang Haken finally prove the 4-color theorem. Here I outline an accessible proof of the simpler 5-color theorem based on Percy Heawood‘s 1890 salvage of Alfred Kempe‘s 1879 flawed proof.

## Convert map to graph

Replace the map with a graph of $V$ vertices, $E$ edges, and $F$ faces. Represent the 5 colors by the numbers 1, 2, 3, 4, 5.

## Euler characteristic

For connected planar graphs, the Euler characteristic $\chi = V-E+F=2$ is a topological invariant. The proof is by induction on the number of vertices. An isolated vertex $V = 1$ connects to no edges $E = 0$ but is surrounded by a single face $F = 1$ extending to infinity, so $\chi = 1-0+1=2$. Assume $\chi = 2$ for a graph of $V$ vertices, and extend it in one of two ways: add an edge and a vertex, so $\Delta \chi =1 - 1 + 0 = 0$, or add an edge connecting two existing vertices, so $\Delta \chi =0 - 1 + 1 = 0$. Either way, the Euler characteristic is invariably $\chi = 2.$

## Twice edges is at least thrice faces

For simple planar graphs (with no loops or repeated edges), double the number of edges is at least triple the number of faces. To prove this, first consider a maximally triangulated graph where each face (including the exterior one) is bounded by 3 edges. Tripling the faces counts each edge twice, so $2E = 3F$. To reach a generic case, deleting any edge reduces this equality’s left side by 2 and reduces its right side by 3, so $2E \ge 3F$.

## At least one vertex has no more than 5 edges

In simple planar graphs, there exists a vertex with 5 or fewer edges. The proof is by contradiction. If each vertex connects to 6 edges, then sextupling the vertices counts each edge twice, so $2E = 6V$. For the generic case, where each vertex connects to at least 6 edges, adding edges increase this equality’s left side without changing its right side, so $2E \ge 6V$ and $E \ge 3V.$

Use this inequality to eliminate vertices $V$ from the tripled Euler characteristic $3\chi = 3V-3E+3F = 6$ and get $E-3E+3F\ge 6$, so $3F - 6\ge 2E \ge 3F$ by the above twice-thrice inequality. But $3F - 6\ge 3F$ implies $- 6\ge 0$, which is a contradiction.

## 5-color theorem

Every map is 5-colorable. The proof is by induction on the number of vertices of the corresponding graphs. A single $V = 1$ vertex is trivially colored by a single color. Assume all $V > 1$ graphs are 5-colorable and consider a $V+1$ graph. Delete the vertex with the fewest edges, which must be 5 or less, and the resulting graph is 5-colorable. Restore the vertex. If it is connected to 4 or less edges, color it differently than its neighbors.

If the restored vertex is connected by 5 edges, consider nonadjacent neighbor vertices $V_1$ and $V_4$ colored by colors 1 and 4. If the subgraph of vertices colored 1 and 4 is disconnected and $V_1$ and $V_4$ are in different components, interchange the 1 and 4 colors in the component including $V_1$ and color the restored vertex 1.

Else if $V_1$ and $V_4$ are in the same component of the subgraph, a Kempe chain barrier separates the subgraph connecting $V_3$ and $V_5$, so  interchange the 3 and 5 colors in one component and color the restored vertex 5.

## Compton Generator

Long before he won the Nobel Prize in Physics, and while still a Wooster undergraduate, Arthur Compton realized a third way to demonstrate Earth’s spin (after pendulums and gyroscopes).

Compton reported his results in a manuscript submitted to the journal Science on 1913 January 13 and published as “A Laboratory Method of Demonstrating the Earth’s Rotation”, Arthur Holly Compton, Physical Laboratory, University of Wooster, Science, 1913 May 23, volume 37, issue 960, pages 803-806.

Compton’s generator is nowadays often used as a text book example of the $\vec F_C = 2 m\, \vec v \times \vec \omega$ Coriolis pseudo-force deflecting the fluid in a circular tube as it is flipped in Earth’s rotating reference frame, which is analogous to the $\vec F_B = q\, \vec v \times \vec B$ magnetic force deflecting a moving charge. However, an inertial perspective is simpler. After viscosity has damped the fluid motion relative to Earth, the fluid farther from the rotation axis is moving faster relative to distant stars, so quickly flipping the ring reverses the speed gradient and induces transient circulation.

Flipping Compton’s ring in a spinning reference frame generates transient fluid flow like flipping a wire loop in a magnetic field generates transient electrical current, where the angular velocity plays the role of the magnetic field; indeed, the former is a motion generator and the latter is an electric generator!

Flipping Compton’s ring π = 180° causes the stationary fluid inside to circulate, generating motion and revealing Earth’s spin.

As in the above figure, assume Earth has radius $R$ and angular speed $\omega$. Assume the ring is at co-latitude $\theta$ and subtends an angle $2\delta$ from Earth’s center. Then the spread in the fluid’s inertial speeds is

$$\delta v = \omega R \sin[ \theta + \delta] - \omega R \sin[\theta - \delta] = 2\omega R \cos \theta\sin \delta.$$

If the ring’s radius $r \ll R$, then

$$1 \gg r / R \approx \delta \approx \sin\delta,$$

and so

$$\delta v = 2 \omega r \cos \theta = 2 \omega r \sin \lambda,$$

where $\lambda$ is the latitude. As checks, $\delta v = 0$ when $\lambda = 0$ (straddling the equator), and $\delta v = 2\omega r$ when $\lambda = \pi/2$ (straddling the north pole).

For Earth’s $\omega = 2\pi/\text{day}$ angular speed, flipping a quiescent $r = 1~\text{m}$ ring at $\lambda = \pi/4$ mid latitude generates an initial fluid speed $\delta v = 0.1~\text{mm/s}$. Compton amplified this motion by using a microscope to view the fluid through a window in a constriction of the tube.

## Analemma

Photograph the sky at the same time each day for a year and Sun will appear to execute a figure-8 path called an analemma, which is often inscribed on Earth globes and can be used as an almanac, as by Tom Hanks‘ character Chuck Nolan in the movie Cast Away.

The 2D animation below illustrates the formation of a 1D back-and-forth analemma for Planet whose spin axis is perpendicular to its orbital plane. Planet’s year divides into 7 equal sidereal days (when red arrow observer points left), 6 equal mean solar days  (when white ray from Sun crosses yellow lines), and 6 unequal apparent solar days (when red arrow points along white ray toward Sun).

Planet’s rotation (spin), which is constant, and eccentric revolution (orbit), which is faster nearer Sun where gravity is stronger, cause mean solar days to be nonuniformly distributed about the orbit, dense near aphelion (far point at left) and sparse at perihelion (near point at right). For an equatorial observer who experiences noon at perihelion, mean noon (when yellow dots record Sun’s direction) precedes apparent noon during the orbital top half and succeeds apparent noon during the orbital bottom half, with the constant rotation “falling behind” during the fast orbital right half and “getting ahead” during the slow orbital left half.

Formation of a back-and-forth analemma (yellow dots) for an un-tilted planet in an eccentric orbit.
(You may need to click to see the animation.)

The 3D animation below (with a wide-angle perspective) illustrates the formation of a 2D figure-8 analemma for an oblique Planet whose spin axis is tilted 45° downward. Planet’s year divides into 40 equal sidereal days, 39 equal mean solar days, and 39 unequal apparent solar days. The constant tilt and elliptical orbit cause an equatorial observer to oscillate above and below Sun. For an equatorial observer who experiences noon at perihelion, mean noon (when yellow dots record Sun’s direction) precedes apparent noon, with Sun above the equator, during the orbital top half and succeeds apparent noon, with Sun below the equator, during the orbital bottom half, forming a figure 8. (In practice, the precise shape and orientation of analemmas vary with time of day and latitude.)

Formation of a figure-8 analemma (yellow dots) for a tilted planet in an eccentric orbit. Showing the planet at mean solar day increments reduces bandwidth but hides most of its spin.
(You may need to click to see the animation.)

## Perseverance, Ignition, Breakeven

Overcoming decades of enormous physics and engineering challenges, and despite persistent pessimism, skepticism, and criticism, the National Ignition Facility has achieved an historic milestone for controlled nuclear fusion, a target energy gain factor of $Q > 1$.

Last week, NIF focussed the world’s most powerful laser pulse on a small gold cylinder that converted the incident ultraviolet light into x-rays and caused an enclosed diamond-coated deuterium-tritium pellet to implode and convert some of its matter to energy: 2.05 MJ of energy went into the target, and 3.15 MJ came out.

I remember the fusion goals of my childhood as ignition and breakeven.  On 2021 August 8, NIF achieved ignition by surpassing the Lawson criterion (roughly, the triple product $\text{density} \times \text{temperature} \times\text{confinement time}$) to create a self-sustaining “burning plasma”, where fusion heating exceeded all cooling processes, but with a target energy gain of only $Q \approx 0.72$. Finally, on 2022 December 5, after 16 months of additional hard work, including increasing and balancing the laser power and thickening the pellet’s diamond coating, NIF achieved both ignition and breakeven with a record target energy gain of $Q \approx 1.5$.

Although nowadays the “ignition” and “breakeven” criteria are often conflated, as in this morning’s formal announcement, by any standard NIF appears to have at last achieved the grand but elusive goal embodied in its name. Like LIGO’s success in detecting gravitational waves, NIF’s success in achieving ignition and breakeven is a tribute to perseverance and diligence in the face of a daunting challenge.

The Artemis 1 mission’s Orion spacecraft has successfully entered and exited a distant retrograde orbit about Moon. DRO is a stable and easily accessible orbit requiring a low velocity change $\Delta V$. In DRO, Earth‘s non-negligible gravity contributes to a 3-body problem that makes the inertial space orbit non-Keplerian: an ellipse centered — not focussed — on Earth.

The attached animation, which I generated by numerically integrating the 3-body motion equations, displays a DRO in reference frames fixed relative to distant stars [left pane] and rotating with Moon about Earth [right pane]. From the north celestial hemisphere, Orion [red] orbits anti-clockwise relative to Earth [cyan], like our solar system’s planets, but clockwise (and hence retrograde) relative to Moon [white].

For Artemis 1, Orion spent almost a week in DRO, completing a half revolution about Moon (and a quarter revolution about Earth), and is currently returning to Earth. The crewed Artemis 2 will use a free return trajectory for safety, and future Artemis missions will use Near Rectilinear Halo Orbits to avoid Moon periodically eclipsing Earth.

Far from Earth [cyan], the Orion spacecraft [red] entered a distant retrograde orbit about Moon [white]. Relative to Earth [left pane] Orion orbits one way, but relative to Moon [right pane] Orion orbits the opposite way. (You may need to click to start the animation.)

From a distant retrograde orbit, a camera at the tip of an Orion solar panel photographs Earth and Moon, 2022 November 28.

## Artemis Is the Sister of Apollo

I stayed up late last night and early this morning to watch the successful uncrewed launch of Artemis 1. In Greek tradition, Artemis was the twin sister of Apollo, and the Artemis program hopes to return humans — including the first woman — to Moon as preparation for sending them onward to Mars.

As a child of the Apollo program, I am convinced that a future with humans living and working in space, on Moon and Mars, is much more exciting than one with humans confined to Earth, and in this regard the half-century gap between Apollo and Artemis has been deeply disappointing.

Today real hope exists for realizing Apollo’s promise to permanently extend the bounds of human experience beyond low-Earth orbit. Later this decade, the crewed Artemis 2 should circumnavigate Moon and the crewed Artemis 3 & 4 should land on Moon’s unexplored South Pole. I hope these and subsequent increasingly ambitious missions and their diverse explorers will excite and inspire a new generation, the Artemis generation; I know they will excite and inspire me.

As currently planned, Artemis 3 & 4 will require the development of a brand-new revolutionary launch vehicle — a true 21st century super-heavy lift rocket, which I have highlighted before and expect to soon discuss again.

A camera attached to the tip of the Orion capsule’s service module photographed Artemis 1 departing Earth orbit on 2022 November 16