Assume there is an infinitely large plane with a charge density $\sigma$. I understand how to derive, using Gauss' Law, that $E = \frac{\sigma}{2\epsilon_0}$ is the electric field at a distance $r$ from the plane.  However, intuitively, I don't understand how this could be true. Shouldn't the electric field be more powerful closer to the plane, since the electric field falls of as $\frac{1}{r^2}$? The larger I make $r$, the further it is from every point, meaning the electric field should decrease - shouldn't it?  Any help understanding this would be appreciated.
For an infinite line charge, $E$ falls off with $1/r$; for an infinite sheet of charge it's independent of r! The infinitesimal contributions to $E$ fall off with $1/r^2$, so why doesn't the total $E$ fall off the same way for the infinite line and sheet charges?
I know how to , but I'd like to add a bunch of different extensions and avoid the tedious job of adding every single one. Is there a command line tool or registry key I can use to accomplish this? How does delete[] "know" the size of the operand array? Foo* set = new Foo[100]; // ... delete [] set;   You don't pass the array's boundaries to delete[]. But where is that information stored? Is it standardised? Determinant of a matrix whose $(i,j)$th element is $|i-j|$? Question states: "Let $D_n$ denote the determinant of the $n$-by-$n$ matrix whose $(i,j)$th element (the element in the $i$th row and $j$th column) is the absolute value of the difference between $i$ and $j$. Show that $D_n = (-1)^{n-1}(n-1)2^{n-2}$."  It seems like that I can get a recurrence relation (especially the question was from a book chapter dealing with recursion). In trying to get a recurrence relation, here is what I have done so far:  (Let $R_k$ denote the $k$th row and let $(i,j)$ denote the element/spot of $i$th row and $j$th spot)  1) replace $R_1$ with $R_1-(R_2+R_{n-1})$ to get $R_1^*$  2) replace $R_1^*$ with $R_1^*+R_{n-2}$  3) If I am not mistaken, I should have $-2$ on $(1,1)$, $2$ on both $(1,n-1)$ and $(1,n)$, and all other element in the first row as $0$.  4) take out $2$ from the first row, and expand the determinant about the first row  5) I should have $D_n=2(-D_{n-1}+(-1)^n B_{n-1})$, where $B_{n-1}$ is the determinant of a not-easily-reducible mayrix whose last column consists of $n-1$ ones followed by one $-1$.  How should I go about from now on? Once I get a recurrence relation, it would be pretty easy to get the explicit formula for $D_n$ using induction.  P.S. This question apparently is a question taken from 1969 Putnam Competition Cyclic vector implies commuting linear operator is a polynomial Let $T$ be a linear operator on the finite dimensional vector space $V$. Suppose $T$ has a cyclic vector. Prove that if $U$ is any linear operator which commutes with $T$, then $U$ is a polynomial in $T$. GPU compute grayed out I have a simple question but I am not sure on how to proceed. I would like to put 10 figures on the same page: a grid of 5 rows and 2 columns.  What should I type in order to have the figure inserted like this and with equal alignement/size ? Eventually with custom captions... ;) Manual/automatic line breaks and text alignment in TikZ nodes How can I insert a line break in a TikZ node? Simply putting \\ where I want the break doesn't work (see MWE).  Is there a way to make lines break automatically at some specified width?  And can I control the alignment of the text (left, right, centered, justified)?  \documentclass{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \node {First line\\second line}; \end{tikzpicture} \end{document} Is the Macbook 12 inch's USB-C charger interchangeable with other USB-C chargers or devices? If there is only the Macbook 12 inch USB-C charger nearby, can it be used to charge other USB-C devices, such as any Android phones that also use USB-C? (will it be "too powerful" and damage other devices?)  Vice versa, if there is a Quick Charge 2.0 or 3.0 USB-C charger from other devices nearby, can it be used to charge the Macbook 12 inch?  The main concern is about whether it could damage the device or battery. How to log commands within a "sudo su -"? If I:   [user@notebook ~] sudo echo 123456uu 123456uu [user@notebook ~]    Then I can see that in the logs:  [root@notebook /var/log] grep 123456uu * auth.log:Jan  9 17:01:51 notebook sudo: user : TTY=pts/3 ; PWD=/home/user ; USER=root ; COMMAND=/bin/echo 123456uu [root@notebook /var/log]    but if I:   [user@notebook ~] sudo su - [root@notebook ~] echo 1234567zz 1234567zz [root@notebook ~]    I cannot see it in the logs:   [root@notebook /var/log] grep 1234567zz * [root@notebook /var/log] echo $? 1 [root@notebook /var/log]    My question: How can I turn the logging on for the commands within the "sudo su -"?  OS is an Ubuntu 12.04 but the question is in general.   UPDATE#1:   [user@notebook ~] sudo su - [sudo] password for user:  [root@notebook ~] echo zizizi zizizi [root@notebook ~] cd /var/log [root@notebook /var/log] grep -iIR 'zizizi' * [root@notebook /var/log] grep 'COMMAND=/bin/su -' * auth.log:Jan 10 15:42:42 notebook sudo: user : TTY=pts/1 ; PWD=/home/user ; USER=root ; COMMAND=/bin/su - [root@notebook /var/log] can anyone help me with this, I got stuck on it.  let $a_n$ and $b_n$ (for $n$ a non-negative integer) be two sequences of real numbers such that $\lim_{n\to \infty}a_n=a$ and $\lim_{n\to \infty}b_n=b$. prove that $\lim_{n\to \infty} \frac{a_0b_n+a_1b_{n-1}+....+a_nb_0}{n}=ab$. \pause in tikzpicture breaks footline I've discovered that you can \pause within a tikzpicture, which works well until I switched the theme to one with a footline.  I've created a "test case" that illustrates the problem.  \documentclass{beamer} \providecommand\thispdfpagelabel[1]{} % Not sure what this does but our installation requires it. \usepackage{tikz} \usetheme{Madrid} % Has a footline.  \begin{document}   \begin{frame}     \frametitle{test}     \begin{tikzpicture}       \node at (0, 1) {Hello};       \pause       \node at (0, 0) {World};     \end{tikzpicture}   \end{frame} \end{document}   This creates two slides, but the footline is only drawn on the second (the headline/title appears on both). Is there anything I can do about this? Why 'mkpasswd -m sha-512' produce different result every time it is called? I am using this function to create a password in the shadow file  $ mkpasswd -m sha-512 password $6$o50n4EfNOBzfs$K5l5AcOI5e0I9n2df0LmR11OLSPKSVM6AGBOnWVfDDfMubb46MrAj0zehdLNEoYzD2fj5q4XUdDwexj/dWLHy/ $ mkpasswd -m sha-512 password $6$ADmDOhDpW$gsYQuiKPpV1ewaUpRq2VGAvrcocffLQ8XGF94e6LNLZfKIjy6Ku.cZoR/5exeCi1ESjPB9TRM6HxCiY5BCNCV0 $ mkpasswd -m sha-512 password $6$Efj1agaI$HGkny3q1OTYT4KREh18gueHJae/3Bvil0iOEhfXj8bD.qy9Lg2UIQJMuBcq0XtG3xzueK.7cp0GfKr7tEo5YI/ $ mkpasswd -m sha-512 password $6$zdVPlCPai4Y$x3CBvlP99xZXZcr4PTiE..YLpZx39h5OHDxqazd9wFLImPuwsXF0M6KmqLzlCCrnQhI2lmEPdCzfmHA/fDiOz.   It makes me wonder how does the output value work as a password Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $ Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... Proving a graph has no Hamiltonian cycle Is there such a thing as charge density in quantum mechanics? Is there such a thing as a time-varying charge density in quantum mechanics? I am looking at the tungsten filament in a light bulb. Am I allowed to say that there is a wave function which describes that filament, and that in some sense there is a time-varying charge density (which some people call a probability) associated with the square of that wave function? There are electrons which have a probability distribution, and there are nuclei which are vibrating within the lattice, but all in all is there, or is there not, a time-varying charge distribution associated with that glowing filament? And if not, why not?  EDIT: There seems to be some question as to just what kind of charge-density fluctuations I am talking about. Let me clarify: in a hydrogen atom in the ground state, I would say there are NO charge density fluctuations. But if there is a superposition of the ground state and a 2p state, I would say the charge distribution fluctuates sinusoidally at the difference frequency of the two states. THAT is the kind of charge density fluctuation I am asking about for in the tungsten filament. Does it or doesnt' it?  I hope this clarifies the intent of my question.   RE-EDIT: In a separate (and rather hard-fought) discussion, it appears that people who know more than me agree that there is indeed an oscillating charge distribution in the hydrogen superposition as described above (see  ).  So if there is an oscillating charge in the hydrogen atom, why wouldn't there be oscillating charge in a heated tungsten filament? When a person lifts a book while standing on the earth's surface. Why is his force considered external to the book-earth system? Isn't his force internal like the case of the forces that cause the exploson of a bomb. I think the forces the person exerts on both the book and the earth are internal since they are equal but opposite therefore resulting in the position of the centre of mass of the system being maintained and if i'm right is the mechanical energy of the system constant?  Am i wrong? Please correct me. Error to calculate date diff with DateTime class If you draw two cards, what is the probability that the second card is a queen? We had this question arise in class today and I still don't understand the answer given. We were to assume that drawing cards are independent events. We were asked what the probability that the second card drawn is a queen if we take two from the deck. The answer given was 4/52, which seems counter-intuitive to me. How is the probability still 4/52 if there was a card drawn before it? What if the first card drawn was a queen? By accident, I recently removed some videos from my phone. I tried to restoring them with Recuva and TuneUp Utilities and the results were "undf" files.  I tried to open with VLC, Media Player Classic and other video players to no avail. Where/how do I find out the codec for these files?   I would really love to be able to watch these videos again. How to prove or disprove statements about sets Prove or disprove:  The set: $\{ \emptyset\}^{\Bbb N}$ has only one element.  The set $\emptyset^{\Bbb N}$ is empty.    I'm pretty sure both are true, for 1. all the natural numbers go to the empty set and that is the only element. Where as 2. all the natural numbers go to nothing so that's why it's empty. (Same as $ {\Bbb N}^{\emptyset} $ nothing goes to something is nothing (I think)). The problem is I have no idea how to show this so advice would be appreciated.  Edit: My defintion of a function is: $f:B\to A$ if  $(a_1,b),(a_2,b)\in f \ \Rightarrow \ a_1=a_2$ So for 2. I can simply say that: $(1,\emptyset),(2,\emptyset)\in f \Rightarrow a_1\neq a_2$  ?