Obviously, with quantum mechanics, we know particles don't really orbit in electric fields.  From what I know, clasically, Bremsstrahlung radiation would occur if an electron was deflected or accelerated around a proton, and the energy of that photon would be equivalent to the energy lost.  But if an electron did classically orbit a proton in a perfectly circular orbit, it would never change energy and could never emit a photon, yet it would be accelerated to form a circular path.  How would the electron emit any radiation if it was not changing energy in a circular path?
If a charge travels in a circle it must accelerate, thereby producing EM.  However, a wire in a circular loop is analogous to many charges moving in a circle. So, why don't circular currents produce EM? (I have not found any evidence that circular currents can produce EM).  A similar question is: . However, my question asks about circular wires and direct current. Thanks.
Smallest dense subset of $\mathbb{R}$ I am not sure if what I am looking for even makes sense (or) exists. Anyway I would be happy if someone can clear my confusion.  The set of real numbers $\mathbb{R}$ is obtained as completion of $\mathbb{Q}$. However, $\mathbb{Q}$ is not the only set which is dense in $\mathbb{R}$. $\mathbb{Q} \backslash \mathbb{Z}$ is also a dense subset of $\mathbb{R}$. I am wondering if it makes sense to talk of the "smallest" dense subset of $\mathbb{R}$. To phrase what I am looking for precisely and what I mean by smallest, I am looking for a dense set $A$ of $\mathbb{R}$ such that if $B$ is a proper subset of $A$ then $B$ is not dense in $\mathbb{R}$. I am able to "see" that the set $A$, I am looking for doesn't exist since any open interval consists of infinite rationals. But I am unable to precisely argue out to myself and convince why $A$ doesn't exist. Could some one throw more light on this? How can I use adb to send a longpress key event? In Ubuntu 18.04, I can create a file as one user in /tmp, and write to it as user root (despite the sticky bit being set on /tmp). In Ubuntu 20.04, I get a Permission Denied error as user root. Ubuntu 18.04: # As user &quot;ubuntu&quot;: touch /tmp/tmplock  # As user &quot;root&quot;: echo &quot;foo&quot; &gt; /tmp/tmplock  cat /tmp/tmplock # Outputs &quot;foo&quot;  ls -lah /tmp drwxrwxrwt 32 root   root    20K Jun 16 18:36 . -rw-rw-r--  1 ubuntu ubuntu    4 Jun 16 18:36 tmplock  Ubuntu 20.04: # As user &quot;ubuntu&quot;: touch /tmp/tmplock  # As user &quot;root&quot;: echo &quot;foo&quot; &gt; /tmp/tmplock # Outputs: bash: /tmp/tmplock: Permission denied  ls -lah /tmp drwxrwxrwt 12 root   root   4.0K Jun 16 18:15 . -rw-rw-r--  1 ubuntu ubuntu    0 Jun 16 18:14 tmplock  I can't find any information on if this is related to the sticky bit, or an underlying Linux kernel change, or anything. Does anyone know what can explain the change in behavior? How to call shell script from php that requires SUDO? Example of $\deg(fg)<\deg(f)+\deg(g)$ Let $R$ be an integral domain and $f,g\in R[X_1,...,X_n]$ where $n&gt;1$. What is an example of a pair $f,g$ such that $\deg(fg)&lt;\deg(f)+\deg(g)$?   Moreover, I have proven that the units of $R[X_1,...,X_n]$ are just the units in $R$. (Proof is done inductively) Is it true?  EDIT    Even though the each term in the underlined summation is nonzero, isn't it possible that the summation is 0? Why is this impossible ? How should I unit test multithreaded code? I have a fresh installation of Ubuntu 14.10 on my laptop. It appears that letters are missing from files and folders, as well as menus and configuration windows. I'm not sure where to start in order to fix this. Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.  I am studying Baire space and I would like to try the following facts :      $(i)$ The inverse image of dense set is dense.       $(ii)$ The inverse image of comeager set is comeager.   I am studying Baire space and I would like to try the following facts  Can we collaborate to verify the above claims?    Thanks. Seeing as C# can't switch on a Type (which I gather wasn't added as a special case because is relationships mean that more than one distinct case might apply), is there a better way to simulate switching on type other than this?  void Foo(object o) {     if (o is A)     {         ((A)o).Hop();     }     else if (o is B)     {         ((B)o).Skip();     }     else     {         throw new ArgumentException("Unexpected type: " + o.GetType());     } } What are the differences between Factor Analysis and Principal Component Analysis? It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use one over the other. A real example would be incredibly useful. Only ftps with proftpd on debian Why is the ROC curve two-dimensional instead of three-dimensional? When I understand it correctly, a  curve composes of a lot of ratios of TPR and FPR. Each such ratio is gained, e.g., by scanning/investigating a parameter where the ratio is the result of the used classifier. In other words: I'm interested in how a classifier separates data points according to a frequency interval of 0-100 Hz. Hence I scan this frequency, record all the 101 ratios, put them into the plot and the ROC curve represents these datapoints, finally.  Wouldn't it be much more interesting when the frequency would be used as a third axis? Or is it implicit in the axis? But anyway, I think it would be easier to look at with a third axis? Is there a word to describe words that can be spelled different ways? Is there a word in English to describe words that can be spelled more than one way?  For example:   colour and color favourite and favorite   However, this shouldn't be limited to words spelled differently in different dialects, though I'm not sure what other case there would be. I'm trying to process some data in Excel. The data includes numeric account numbers and other long numeric strings (like cell phone MEIDs). I am not doing math operations on these strings, so I want Excel to treat them as plain text.  Here is what is making me nuts (this is Excel 2010):   Take a long number like 1240800388917 and paste it in a cell in a new worksheet. Excel's default cell format is general, so the string is presented in scientific notation as 1.2408E+12 Right click on the cell, select Format Cells, set the format to Text   The cell is still displayed in scientific notation, even though the format has been set to text.  Now, if I do the steps in a different order:   Format an empty cell as text. Right click on the cell, select Format Cells, set the format to Text Take a long number like 1240800388917 and paste it in to the text formatted cell   Now, the cell is displayed as a string and not in scientific notation.  The results remaining in scientific notation even though the cell is formatted as text just seems broken to me. I've seen suggested work-arounds like: use CSV import and set the format to text, add a space character to the beginning of each numeric string, and others.  Is there a simple good work around to easily keep these strings formatted as text?  Why on earth does Excel do this?  Related SU questions I found:  and Hydrostatic pressure - doesn't density vary with depth? Our class is learning about hydrostatic water pressure and we have been told that we can calculate the force of the liquid on an object at any depth using "the density x 9.8 x the depth". However, as the depth increases, wouldn't the density of the liquid increase because of the weight of the liquid above it compressing it? So should't there be something in the equation to account for the varying density? To me, "density x 9.8 x depth" seems like it is saying that the density will be constant... Limit of sequence in which each term is defined by the average of preceding two terms We have a sequence of numbers $x_n$ determined by the equality   $$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$  The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$ $$\lim_{n\rightarrow\infty} x_n $$  The options are:   A)$\frac{x_0 + 2x_1}{3}$ B)$\frac{2x_0 + 2x_1}{3}$  C)$\frac{2x_0 + 3x_1}{3}$ D)$\frac{2x_0 - 3x_1}{3}$  Since it was a multiple choice exam I plugged $x_0=1$ and $x_1=1$. Which means that all terms of this sequence is $1$,i.e, $$x_n=1, n\in \mathbb{N} $$  From this I concluded that option A was correct.I could not find any way to solve this one hence I resorted to this trick. What is the actual method to find the sequence's limit? Get Label value for specified language When do mobs despawn in Minecraft? I understand monsters only spawn in a 9 x 9 grid of chunks centered on the player, and I assume they stay around while they remain within that distance of you. But do they despawn as soon as you move away from them? Or do they remain? If so what (if anything) causes them to despawn? Are the rules any different for passive mobs?  Will running far away from an area and then coming right back clear it of monsters? How can I make Firefox download different file types to different locations? 98% of what I download is music, so I set the download location to my music folder. However, the other 2% has no place there, so every time I download .doc files or PDF files I have to remove them. Is there a way to make Firefox change the download folder based on file extension? I'll settle for all non-MP3 files getting saved to the desktop.  I know I can right click, choose Save Link As and pick the download location myself, but it seems like there should be a better way. When I use any variation of English, US international (with dead keys,altGr dead keys or alternative) on my Linux Mint machine I always encounter this behaviour. When I press one of these keys: ' " and then follow them with a 'non-accentable' character like a [ or a b no output comes out at all.    Whereas in Windows US-International it would print [ or b. If I wanted to type this I would have to escape each dead key with a space instead of with any 'non-accentable' character. This is annoying when programming (not really, but I trained with the Windows 'Qwerty International' on typing.io and switching back and forth between the systems is irritating).    Is there any way to change that so it works like in Windows?