I understand Current density $J =n e  v$ [$n=$ carrier concentration per unit volume, $e=$ charge on electron, $v=$ drift velocity] Which is independent of area of cross section of wire (and hence the diameter) However, the book practical electronics for beginners quotes that:  Current density within a copper wire increase as the diameter of the wire decrease.  Is the book mentioning about varying cross section in the same wire? If yes then can you please explain how does current density increase in a smaller diameter wire?
I am confused whether drift velocity depends upon area or not . I checked answers for this question but all of them were cancelling each other out . from the equation $$I = neAv$$ It is clear that drift velocity should depend upon area but at the same time don't you think that by increasing the area of cross section of a conductor should have resulted in increase in drift velocity please help me out with it in detailed answer of how should i go with all the possible cases and explanation ,thanks :)
"Above"/"below" before/after a noun I have seen sentences similar to the following:  (1) See the reference above. (2) See the reference below.  And,  (3) See the above reference.  But not,  (4) See the below reference.  Are all these forms acceptable? Which is/are preferred in formal writing? How do I survive in the Nether? In the course of creating the content for a site, we've ended up with orphaned content nodes that are not linked to from anywhere. I'd like to find and examine these. Surprisingly, I can't find a module for this.  Tiny hypothetical example: I'm building a brochureware site. Someone wrote a "Contact" page, someone else wrote an "About Us", and someone else wrote "Directions". The person writing "Directions" forgot to put it in a menu link or add a link from "About Us" to "Directions". So "Directions" is orphaned: there are no links to it. I'd like to find all such nodes so I can figure out what to do with them. What is the perception of students who take a gap year between their graduation and starting graduate school? I am an undergraduate mathematics major, entering my fourth year. I plan on applying to graduate school, but I have not yet taken the GRE exams, and there a few key courses (Topology and Algebra in particular) that I have not taken yet, so I am hesitant to take the Advanced Math GRE too soon. (If you're wondering why I haven't taken Topology and Algebra yet, it's because I was trying to secure a CS degree, which sidetracked my mathematical studies last year. It was a healthy branching out experience, and one that helped me realize my true academic passion.)  My main question: what is the perception of students who take a gap year between their graduation and starting graduate school?  I think I will take the GRE tests this fall, a) for practice and b) if I score high, then I might as well apply sooner rather than later. However, I ask this question to gauge how much energy I should put into preparing for the tests this fall. If I hear, "You should really do your best to avoid any gap in your education," then I will work my butt off in preparation for the tests. But if I hear, "It's totally okay take a gap year before going into graduate school," then I would give myself more freedom in terms of my non-standardized-exam studies, extracurriculars, and in general social activities. I figure the answer to this question won't be as cut-and-dry as I hope, but any advice would be appreciated!  Just to clarify, my gap year would not consist of me sitting at my desk taking practice GRE exams all day... That would be a pitiful use of my time. I definitely plan working either as an intern or a research assistant at a successful organization. It seems like the general consensus is that taking time off is okay, and potentially beneficial, assuming you actually do something in that gap. How to read a line from the console in C? What is the simplest way to read a full line in a C console program The text entered might have a variable length and we can't make any assumption about its content. Sudden increase in timing when summing over 250 entries How do you plot time on the X axis in excel? I need to plot the decay of my phone's battery against time over night. My starting time is 23:17:47 and my ending one is 07:05:49. I don't have any issue in changing the time format in the current one, selecting an XY chart and plotting time on the X axis.   The problem arises because Excel doesn't start the X axis at 23:17:47, but instead at 00:00:00, so my first value is drawn from the end of the graph and the goes back at the beginning when time values are over 00:00:00.   In other words, how do I set up the min time value to 23:17:47 and the max to 07:05:49 in excel? Does anyone know the (specific) reasons that it is recommended that old LTS server users wait until the point release to upgrade?  Is it just a matter of bug fixes and waiting for improved stability for production servers, or is there something more fundamental than that? Is there any way you could revive Guest Sessions in a future LTS distro?  Guest sessions are good security - they give me just enough power to do what I need to do on a daily basis without risking my system. In short, Guest sessions save me from myself.  I had upgraded from 16.04 to 18.04, gotten a problem I couldn't fix and was all to happy to wipe and rebuild my hard drive and return to 16.04.  I can't be the only one who feels this way. How can I convert monsters from D&D 3 or 3.5 to 2ed monsters? Is it possible to know who voted my question/answer up or down? If not, then is it possible to include this feature? By this feature we can come to know if you are being voted up by question owner or persons giving other answers to same question or just the visitors of that question thread.  Like...  If it is a group of people giving other answers, then your answer is most correct If it is readers of that question, then your answer is more popular I've heard that in the study of finite fields, and other concepts related to finite fields, mathematicians have found a sort of gap: there are various results and things that seem like they correspond to a field with one element, $\mathbf{F}_1$, even though there is no such field. Of course, there is an object which seems similar to a field with one element: the zero ring, $\mathbf{Z}_1$, defined as having one element, $0$, obeying the equations $$0 + 0 = 0 - 0 = 0 \cdot 0 = 1 = 0.$$ The zero ring seems to behave like a field in every respect, except for the fact that it fails to satisfy $0 \ne 1$. (But the importance of the axiom $0 \ne 1$ isn't clear to me.) However, in any case, there seems to be consensus that the zero ring is not the field with one element. People say that the zero ring &quot;does not behave like a finite field&quot; () or that it &quot;does not have the features that mathematicians need&quot; (). I'm not familiar enough with algebraic geometry to understand the properties that $\mathbf{F}_1$ is expected to have. Is there an elementary, undergraduate-level explanation of  why it seems like a field with one element &quot;ought to exist&quot; in the first place, what properties we expect it to have and why, and how the zero ring fails to satisfy these properties?  Or do I need to study algebraic geometry if I want to have an understanding of any of this? (A side question: do we have a good answer to the question of whether the single element of $\mathbf{F}_1$ ought to be $0$, $1$, both, or neither?) Between vs. Among - "Difference [between | among] one thing and other things." Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a&gt;1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition.   proof:  given $a_n=a^{\frac{1}{n}}$ for $a\in\mathbb{R},a&gt;1$.  for $n\in\mathbb{N}^*,n+1&gt;n\Rightarrow \frac{1}{n+1}&lt;\frac{1}{n}$ and because $a&gt;1$ we gets $a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}$, since $\frac{1}{n+1}&gt;0$, then $1=a^{0}&lt;a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}\Rightarrow 1&lt;a_{n+1}&lt;a_{n}$.  then we proof that $\lim\limits_{n\to+\infty}a_n=1$ we need to show that $\forall\epsilon&gt;0,\exists N,\forall n&gt;N,|a_n-1|&lt;\epsilon$  then for $\epsilon&gt;0$, choose $N=\frac{1}{\log_a(\epsilon+1)}$, since $a&gt;1$ we gets that $$\forall n&gt;N,1&lt;a^{\frac{1}{n}}&lt;a^\frac{1}{N}\Rightarrow0&lt;a^{\frac{1}{n}}-1&lt;a^{\frac{1}{N}}-1\Rightarrow |a^{\frac{1}{n}}-1|&lt;|a^{\frac{1}{N}}-1|=|a^{\log_a(\epsilon+1)}-1|=|\epsilon+1-1|=\epsilon\Rightarrow |a^{\frac{1}{n}}-1|&lt;\epsilon$$  wich implies that $\lim\limits_{n\to+\infty}a_n=1\square$  my proof to the limit is correct? Number of planks required to stop the bullet TikZ: remove margin, padding, border around nodes containing images I want to create a simple figure where images are connected with arrows. Here is what I did:  \begin{tikzpicture} \node[](A) {\includegraphics[width=0.2\textwidth]{a}}; \node[right=1mm of A](B) {\includegraphics[width=0.2\textwidth]{b}}; \draw[-&gt;] (A) to (B); \end{tikzpicture}     But I want the arrow to start and end right at the images with no space between. How can I do that? How do you sort a dictionary by value? I often have to sort a dictionary, consisting of keys &amp; values, by value. For example, I have a hash of words and respective frequencies, that I want to order by frequency.  There is a SortedList which is good for a single value (say frequency), that I want to map it back to the word.   orders by key, not value. Some resort to a , but is there a cleaner way? How to avoid messages from applications which I ran with a &? I ran dolphin like this:  dolphin . &amp;   I see messages in the console like these:  QPixmap::scaled: Pixmap is a null pixmap   How can I keep the console clean or silent? How do I save/remember last-used window position and size for applications? I'm interested in the exact barrier of $$\ln(n!)= \Theta(n\ln(n))$$  and if it even exists.  This means, there is a $c_1, c_2$ , so that $$\ln(n!) \le  \Theta(n\ln(n) c_1$$  $$\ln(n!) \ge  \Theta(n\ln(n) c_2$$  or  $$0 \le \lim\limits_{n \to \infty}\frac{\ln(n!)}{n\ln(n)} \ge \infty$$  I could use de l'Hospital for the limit fraction, since it's $\frac{\infty}{\infty}$ but the derivate of $\ln(n!)$ causes problems. Also I don't know how I could prove the two statements above or how to prove them wrong. Any help is appreciated.