electrostatics - Does Coulomb's Law, with Gauss's Law, imply the existence of only three spatial dimensions?

Coulomb's Law states that the fall-off of the strength of the electrostatic force is inversely proportional to the distance squared of the charges.

Gauss's law implies that a the total flux through a surface completely enclosing a charge is proportional to the total amount of charge.

If we imagine a two-dimensional world of people who knew Gauss's law, they would imagine a surface completely enclosing a charge as a flat circle around the charge. Integrating the flux, they would find that the electrostatic force should be inversely proportional to the distance of the charges, if Gauss's law were true in a two-dimensional world.

However, if they observed a $\frac{1}{r^2}$ fall-off, this implies a two-dimensional world is not all there is.

Is this argument correct? Does the $\frac{1}{r^2}$ fall-off imply that there are only three spatial dimensions we live in?

I want to make sure this is right before I tell this to my friends and they laugh at me.

Answer

Yes, absolutely. In fact, Gauss's law is generally considered to be the fundamental law, and Coulomb's law is simply a consequence of it (and of the Lorentz force law).

You can actually simulate a 2D world by using a line charge instead of a point charge, and taking a cross section perpendicular to the line. In this case, you find that the force (or electric field) is proportional to 1/r, not 1/r^2, so Gauss's law is still perfectly valid.

I believe the same conclusion can be made from experiments performed in graphene sheets and the like, which are even better simulations of a true 2D universe, but I don't know of a specific reference to cite for that.

mass - Redefining the kilogram using Planck's constant instead of the density of water among other examples

The kilogram is in the process of being redefined in terms of Planck's constant so as to eliminate its dependence on a physical artefact. Since the length and temperature units are already precisely defined, why not just calculate the density of some substance, say water, at a particular temperature and use that as a standard for mass? Sounds simpler to me.

Answer

Since the length and temperature units are already precisely defined, why not just calculate the density of some substance, say water, at a particular temperature and use that as a standard for mass?

Water is a lousy choice. The initial proposal for the French metric system used the mass of a cubic decimeter of water. Measurement issues resulted in this being changed to a prototype-based system in just a few years. Issues with those initial prototypes resulted in the current prototype masses. Issues with those newer prototypes are part of what motivated the physics-based redefinition of the International System (SI).

Water is a bad choice, but what about some other substance? The problem with this is that it flies in the face of one of the key goals of the proposed redefinition of the SI, which is to define the base units of time, length, mass, current, and temperature solely in terms of fundamental physical constants. The mole is also being redefined, from the number of atoms in 12 grams of ¹²C to a specified number.

Other key goals are that the changes should represent improvements and that the redefined base units must be consistent with the past. Those latter two have always been goals. Using a fundamental physics-based approach is new, or almost new. The definitions of the second and meter are fundamentally-based. The improvements that these redefinitions enabled were a strong motivator to continue this process to the remaining three physical units, and to the mole as well.

That said, using a carefully measured quantity of some substance is close to one of the two approaches being used to establish the exact value of Planck's constant. Those two approaches are the Kibble balance (formerly Watt balance), which carefully compares electrical power to mechanical power, and the Avogadro technique, which carefully calculates the number of atoms in a carefully measured sphere of nearly pure ²⁸Si.

The deadline for measurements by multiple groups using these two techniques to estimate Planck's constant passed on July 1. The requirement by the International Bureau of Weights and Measures (BIPM; the acronym is French) was to have at least three experiments with an uncertainty of 50 parts per billion (ppb) or better and at least one with an uncertainty of 20 ppb or better. Previous failures to meet that goal is the key reason the SI redefinition is not yet in place. That goal has now been met. There are multiple experiments with much better than the requisite 50 ppb uncertainty and three with better than 20 ppb.

quantum field theory - Doing a Gaussian Integral

When you integrate over p you get:

by using

What are the steps to this? Do you integrate by parts?.

Find the shortest chess game with 18 queens?

From the *starting position of a chessboard, you need to move the pieces to end up with 18 Queens following the chess rules at all times (this includes white moves first). The solution is the series of moves (not the final position, as there are plenty).

Having other remaining pieces is possible and allowed (e.g. a remaining bishop + the 18 queens and 2 kings).

_{* starting position FEN: rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1}

_{see also the orignial question on chess stackexchange}

Answer

This is known as the "18-Queens Problem". I found this well-known solution by Friedrich Burchard & Friedrich Hariuc (1976) in 96 Half-moves. I can't make a claim to its optimality, but by looking at it and seeing that no better can be found, I'd say it may well be optimal.

1.e4 f5 2.e5 Nf6 3.exf6 e5 4.g4 e4 5.Ne2 e3 6.Ng3 e2 7.h4 f4 8.h5 fxg3
9.h6 g5 10.Rh4 gxh4 11.g5 g2 12.g6 Bg7 13.hxg7 g1=Q 14.f4 h3 15.f5 h2
16.b4 a5 17.b5 a4 18.b6 a3 19.Bb2 Ra7 20.bxa7 axb2 21.a4 b5 22.a5 b4
23.a6 b3 24.c4 h1=Q 25.c5 h5 26.c6 Bb7 27.cxb7 c5 28.d4 c4 29.d5 Nc6
30.dxc6 c3 31.c7 c2 32.c8=Q c1=Q 33.b8=Q Qcc7 34.a8=Q d5 35.a7 d4
36.Nc3 dxc3 37.Qa6 c2 38.Qa8b7 c1=Q 39.a8=Q Qhd5 40.gxh8=Q+ Kd7 41.g7

bxa1=Q 42.g8=Q b2 43.f7 b1=Q 44.f8=Q h4 45.f6 h3 46.f7 h2 47.Qfa3 h1=Q
48.f8=Q exf1=Q+

I used the same PGN viewer as OP for consistency.

general relativity - Why are orbits around black holes stable?

Black hole theory involves space (or space-time), itself, being sucked into the black-hole, with the event horizon marking the point at which space/space-time is moving faster than the speed of light. I find it really hard to picture how this could be happening while objects maintained a reasonable stable orbit around black holes. If we take the stars that orbit the super massive black hole at the centre of the Milky Way, the orbital dynamics are used to calculate the mass of the black hole, in the normal way. In other words not taking account of the fact space is rushing at some speed inwards toward the black-hole. I appreciate I'm missing some knowledge here. That's the motivation for asking the question.

electromagnetism - Does a current carrying wire produce electric field outside?

In the modern electromagnetism textbooks, electric fields in the presence of stationary currents are assumed to be conservative,$$ \nabla \times E~=~0 ~.$$ Using this we get$$ E_{||}^{\text{out}}~=~E_{||}^{\text{in}} ~,$$which means we have the same amount of electric field just outside of the wire!

Is this correct? Is there any experimental proof?

Answer

Outside a current carrying conductor, there is, in fact, an electric field. This is discussed for example, in "Surface charges on circuit wires and resistors play three roles" by J. D. Jackson, in American Journal of Physics – July 1996 – Volume 64, Issue 7, pp. 855.

To quote Norris W. Preyer quoting Jackson:

Jackson describes the three roles of surface charges in circuits:

1. to maintain the potential around the circuit,


2. to provide the electric field in the space around the circuit,



3. and to assure the confined flow of current.

Experimental verification was provided by Jefimenko several decades ago. A modern experimental demonstration is provided by Rebecca Jacobs, Alex de Salazar, and Antonio Nassar, in their article "New experimental method of visualizing the electric field due to surface charges on circuit elements", in American Journal of Physics – December 2010 – Volume 78, Issue 12, pp. 1432.

experimental physics - How to measure the mass of the electron?

I've done a little bit of research and it seems Millikan was able to measure the ratio between the charge of the electron and its mass. But how can one measure one of the two constants to get the value of the other?

Answer

The mass-to-charge ratio $m/e$ of the electron was first measured by J.J. Thomson, the discoverer of the electron, using cathode rays in 1897:

It should not be surprising that one may measure this ratio even without isolating "individual electrons" because the electric force acting on a charge may be written as $$ F = ma = Ee, \quad a = E\cdot \frac em $$ So what was left was just to measure the mass or charge separately. Millikan and Fletcher did the relevant oil drop experiment in 1909.

The electric force $F=Ee$ acting on a single drop with charge $e$, a single extra (or deficit) electron, may be calculated when it is set equal to the drag force from hydrodynamics, $6\pi e\eta v_1$. The viscosity $\eta$ is the most difficult thing to know but otherwise all quantities are known so $e$ may be calculated.

If one knows the charge and the ratio, one may calculate the mass as $m = e/ (e/m)$.