Hot answers tagged

128

Why is mains frequency 50Hz and not 500 or 5? Engine efficiency, rotational stress, flicker, the skin effect, and the limitations of 19th century material engineering. 50Hz corresponds to 3000 RPM. That range is a convenient, efficient speed for the steam turbine engines which power most generators and thus avoids a lot of extra gearing. 3000 RPM is ...


44

The division is conventionally made at the boundary between where stars end their lives as white dwarf stars and where more massive stars will end their lives in core collapse supernovae. The boundary is set both empirically, by observations of white dwarfs in star clusters, where their initial masses can be estimated, and also using theoretical models. ...


33

It's not a mistake, and conventional current is not wrong or backwards. The labeling of one polarity of charge as "positive" and the other as "negative" is totally arbitrary. It could be done either way and everything would still work out the same. Franklin didn't choose wrong; he just chose. Labeling protons as negative and electrons as positive wouldn't ...


32

Typically: $\rm d$ denotes the total derivative (sometimes called the exact differential):$$\frac{{\rm d}}{{\rm d}t}f(x,t)=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}\frac{{\rm d}x}{{\rm d}t}$$This is also sometimes denoted via $$\frac{Df}{Dt},\,D_tf$$ $\partial$ represents the partial derivative (derivative of $f(x,y)$ with respect to $x$ ...


29

It's a weird quirk of the SI system that the base unit of mass is the kilogram, not the gram. So you'll see a lot of things expressed in kilograms. Of course, scientists in a given field tend to standardize on certain choices of units without any regard to the SI recommendations. And this makes sense; the units you use should be the ones that make your ...


26

In the end, the choice of a single specific number comes from the necessity to standardize. However, we can make some physical observations to understand why that final choice had to fall in a certain range. Frequency Why a standard? First of all, why do we even need a standard? Can't individual appliances convert the incoming electricity to whatever ...


22

The symbol $\Delta$ refers to a finite variation or change of a quantity – by finite, I mean one that is not infinitely small. The symbols $d,\delta$ refer to infinitesimal variations or numerators and denominators of derivatives. The difference between $d$ and $\delta$ is that $dX$ is only used if $X$ without the $d$ is an actual quantity that may be ...


20

As you have expected, there is no sharp divide between the groups. The divide is man made. Since all conductors have some resistance, (except superconductors - follow this link to find out more) and all insulators will conduct some current if they are forced to, this means there is no absolute dividing line between conductors and insulators. Since ...


18

I had an extensive look around, and I turned up four conventions. This included a short poll of google, other questions on this and other sites, and multiple standards documents. (I make no claim of exhaustiveness or infallibility, by the way.) Using $[q]$ to denote commensurability as an equivalence relation. That is, if $q$ and $p$ have the same ...


17

The wording of the question suggests that the electrons were the first objects or particles whose charge required the people to establish the sign convention. But that's obviously not the case. The electron was discovered by J. J. Thomson in 1897 but for much more than a century before that moment, people had already been studying electric (and magnetic) ...


15

(Someone resurrected this oldie in the queue, so just to be a contrary voice...) Ben Franklin did get it wrong. He had just developed a remarkable new theory of electricity in which positive (+) and negative (-) had specific and accurate meanings, and he was unable to apply the two labels in the way he intended. In Franklin's time electricity was thought ...


14

First, recall what a partition is. A partition of a set $X$ is a way to write $X$ as a disjoint union of subsets: $X=\coprod_i X_i$, $X_i\cap X_j=\emptyset$ for $i\neq j$. When the elements of the set $X$ are considered undistinguishable, what matters are the cardinals of the set only, and we have a partition of an integer number, $n=n_1+\ldots+n_k$. For ...


14

It's very common to get mixed up about signs. The only recommendation I can give is to establish a clear sign convention and stick carefully to it. To show what I mean let's consider your skater: I'm going to use the convention that positive is to the right and negative is to the left. remember that quantities like velocity and acceleration are vectors, ...


13

Well, that equation for the force due to electric charges is only true for a very special choice for the unit of the electric charge. Typically, you would write down Coulomb's law as $k\frac{q_{1}q_{2}}{r^{2}}$, where $k$ is a constant of proportionality chosen to make the units work out. IN the SI system, the unit of charge is the Coulomb (C) and the ...


12

There are two separate issues here. (1) Why does it make sense to consider a dipole moment as a vector? (2) Given that it's a vector, why does it make sense to say that it points in this particular direction, rather than the opposite direction. Intuitively, it makes sense to define a dipole as a vector because when we put it in a field, it aligns itself ...


12

The two other answers address the frequency issue. The voltage issue is much simpler. If the voltage is too high, you run the risk of arcs between conductors. The minimum distance between conductors before an arc appears is proportional to voltage. At 240V, you arc at a distance of a few millimeters in air, depending on humidity. More voltage gets clearly ...


11

Yes, to some extent. Once you choose which of the electron or positron is to be considered the normal particle, then that fixes your choice for the other leptons, because of neutrino mixing. Similarly, choosing one quark to be the normal particle fixes the choice for the other flavors and colors of quarks. But I can't think of a reason within the standard ...


10

The relative sign is not just a convention. Once you decide that $E$ is represented by $i\hbar \partial/\partial t$, there must be a minus sign in the formula for $p$, namely $p=-i\hbar \partial / \partial x$. Or vice versa. First of all, there has to be $i$ or $-i$ in all the formulae because $\partial/\partial x$ is an anti-Hermitian operator (because of ...


9

The kilogram is defined by a prototype (the "International Prototype Kilogram", IPK) -- basically, a kilogram is by definition the mass of a metal cylinder sitting in a vault in Paris. People have made a bunch of other metal blocks with almost exactly the same mass (as near as they could get), called "sister copies". To measure a mass extremely accurately in ...


9

You can absolutely have negative absolute pressure in solids or liquids. Think of an elastic solid being forced to expand due to adhesion to the walls of some chamber. That has negative pressure even if the comparison is a total vacuum. Depending on the bulk modulus of the material being stretched and the strength of the interaction with the walls of the ...


9

We introduce a minus sign to equate the mathematical concept of a potential with the physical concept of potential energy. Take the gravitational field, for example, which we approximate as being constant near the surface of Earth. The force field can then be described by $\vec{F}(x,y,z)=-mg\hat{e_z}$, taking the up/down direction to be the $z$ direction. ...


8

The clockwise direction is normally defined by the right hand grip rule. When your thumb is pointing away from you, your fingers are curled clockwise. So when you look at a clock the axis of rotation is away from you through the clock. I'd guess the downvotes are because people believe your question is not physics related, but in fact this rule is how ...


8

The factor of $1/2\pi$ is an artifact of the normalization convention being used for the momentum eigenstates. To begin to see how this is so, let us note that the choice of normalization of a Dirac-orthogonal continuous basis completely determines the form of the resolution of the identity. Writing an arbitrary state $|\psi\rangle$ in a given ...


8

Perhaps some additional information is in order to shed additional light... The whole discussion begs the question: If $\hbar$ is so convenient, why do we have $h$ around? As usual, "historical reasons". Planck originally invented $h$ as a proportionality constant. The problem he was solving was blackbody radiation, for which the experimental data came ...


8

The identification of units with a standard measure happens at the anthropic level: distances, times etc units are developed from the everyday usage, thus convention is the main answer. When a discipline becomes mathematical and is concentrating at a special power of the unit, the use of scientific "nick names" saves times and mental effort for people ...


8

Gigagrams and megagrams are completely valid names. But if you are often using the base unit of kilograms, it's not obvious how many of those you have. It is (by the rules of the BIPM) incorrect to use any additional prefix with kilograms since it already has a prefix. $10^3kg$ may be rendered as a megagram, but (I think) with a loss of easily noting ...


8

$\rho_\psi$, the density matrix, is not an observable/operator evolving in the sense of the Heisenberg equation of motion $$ \mathrm{i}\hbar\frac{\mathrm{d}}{\mathrm{d}t} A = - [H,A]$$ since it is defined, as you correctly write, as a projector on states, hence it is time-dependent in the Schrödinger picture (since there the states it projects on are ...


8

Another way of distinguishing conductors and non-conductors or insulators is with band gap - for good conductors the fermi level of electrons is inside a band - semiconductors have a small band gap and good insulators have large band gaps... Electrons in solids lie in energy bands, whereas in atoms and molecules they have generally sharp levels. If you ...


7

Central to this discussion is a common and widespread misconception: the wrong idea that electric current is a flow of electrons, and that flows of positive charge are impossible. Nope and nope. Only in metals is an electric current a flow of electrons. Protons can flow, and so can ions of both polarities. The type of charge-carrier depends on the type ...


7

This is analogous to the definition of an empty product in mathematics. For a finite non-empty set $S=\{s_1,\ldots,s_n\}$, the product over $S$ can be defined as $$\prod_{s\in S}s=s_1\times \cdots\times s_n.$$ For such a product you'd want disjoint unions to map into products: if $R\cap S=\emptyset$, then you want $\prod_{x\in R\cup S}x=\left(\prod_{s\in ...



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