I know that induced electric field lines form closed loops but we always see that in textbooks, induced electric field lines are drawn circular loops. So my question is that are induced electric field lines always circular in nature or can they be of any shape, and I also want to know why everyone draws these lines as circular?
Whenever we have a changing magnetic flux through a conductor an induced electric field is generated. It is often said that these induced electric field lines are circular in shape.But why is this so?Why are these field lines not elliptical for example? How can this shape be mathematically derived?  Also does this have anything to do with equipotential surfaces?Do such non-conservative electric fields even produce a potential field?If not why?
I have a sword that currently rotates around a pivot point thats off of the actual object. Im new to blender and im not sure how i would go about changing it so that the sword rotates around the handle. Any help would be appreciated! Is "mens" a valid word? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? What is an intuitive explanation of the Expectation Maximization technique? Is there a black hole in the centre of the Milky Way? XeLaTeX update [20201004] and item ball in Beamer Beamer item balls go missing while compiling in XeLateX since yesterday's TeXlive update, however, the same code works with pdfLaTeX. Any solution, please. Here is my MWE: \documentclass{beamer}  \setbeamertemplate{itemize item}[ball] \setbeamertemplate{enumerate item}[ball]  \begin{document}  \begin{frame} \frametitle{A tested title} text \begin{enumerate}     \item this is an item \end{enumerate} text3 \begin{itemize}     \item this is an item \end{itemize} \end{frame} \end{document} Left/Right across multi-line equation I have a multi-line equation that I want to have matching sized parentheses. The advice I read online was to use \left( equation \right. \\ left. rest-of-equation \right) . However, this syntax leaves first half of the equation unaware of what is in the second half. For example, if you include a square root term after the break, the second parenthesis will be larger than the opening parenthesis. When you are using the aligned environment to split a long equation this can look somewhat offputting.    Compare the top equation to the bottom one. The parentheses match in size for the bottom equation, but the left hand parentheses for the top one is  noticeably smaller.  Top Code:   \begin{aligned}[t]   \gamma_{\text{u}} + \frac{1}{108}   \left(57\gamma_{\text{er}} \right. &amp;+  38\gamma_{\text{se}}\\ &amp; {}\pm  \left.\sqrt{2601 \gamma_{\text{er}}^2 + 3648  \gamma_{\text{er}}\gamma_{\text{se}} + 1444 \gamma_{\text{se}}^2}  \right)   \end{aligned}   Bottom Code:  \begin{aligned}[t] \gamma_{\text{u}} + \frac{1}{16} \left(9\gamma_{\text{er}}  +  6\gamma_{\text{se}} \pm  \sqrt{49 \gamma_{\text{er}}^2 + 76 \gamma_{\text{er}}\gamma_{\text{se}} + 36 \gamma_{\text{se}}^2} \right) \end{aligned} 2/15/18 compiz update broke unity This  forced removal of unity and ubuntu-desktop on my 16.04 Ubuntu system.   The offending package that is missing, but unity needs, seems to be compiz-core-abiversion-20151010. It seems to be a , and I can't find it to download.    As it happens, I mostly use LXDE on this system, but I'd be very unhappy if I used Unity...  Since I do occasionally use the Unity desktop, is there a fix for this?  Edit: related questions are rolling in. Looks like a major quality escape          Edit 2: Write access without read access Is it possible for a user to have a write access to a file and not be able to read it? How is it possible?   I tried the following commands:  debianbox@debian:~/posix/io$ touch filetest debianbox@debian:~/posix/io$ ls -l filetest -rw-r--r-- 1 debianbox debianbox 0 14 oct.  03:10 filetest  debianbox@debian:~/posix/io$ echo "Hello World" &gt; filetest debianbox@debian:~/posix/io$ cat filetest Hello World  debianbox@debian:~/posix/io$ chmod u-r filetest debianbox@debian:~/posix/io$ cat filetest cat: filetest: Permission forbidden  debianbox@debian:~/posix/io$   As you can see here, I have write but not read access on this file. How can this be possible? Is this considered as a bug? If not, in what situation would this be useful? Can negative levels get rid of racial HD? What happens when one is affected by something such as energy drain, and permanently loses levels when one only has racial HD? Does the victim of the spell die? Or do they permanently lose racial HD? Overheard on an elevator today,     I didn't realize it was him.   Corrected by the know-it-all,     He.  "I didn't realize it was he."   The know-it-all then went on a rant about how everybody is a dolt for not knowing that.  I swear I have never heard this before in my life.  Is the know-it-all correct?    If so, why is "I didn't realize it was he," grammatically correct?  What rule makes "him" incorrect here? How many fixed points in a permutation I am taking a probability course and we have been of course learning the classic problems of men with hats or variations of that in some other form. The problems being of the type if $n$ men have $n$ hats and mix them all together, whats the probability $0, 1, 2,$ or $k$ men choose their hat.   But from what I can gather is that they are all problems of the same type, for any given random permutation of $k$ objects how many fixed points are there. My professor taught us to solve this using conditional probabilities (conditioning on whether the first person matched and then generating a system of equations) and using the inclusion-exclusion principle. But is there any other way to reason this? Create longer arrows I have a doubt on something. I have a theorem that says that if $R$ is a ring, then $R/I$ is a domain iff $I$ is prime. Since $X^2-1$ is note prime, $\mathbb Z[X]/(X^2-1)$ is not a domain. But, $X^2-1=(X-1)(X+1)$, and since $X-1$ and $X+1$ are prime in $\mathbb Z[X]$, there are coprime, and thus, by the chinese remainder $$\mathbb Z[X]/(X^2-1)\cong \mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1),$$ since $\mathbb Z[X]/(X-1)\times \mathbb Z[X]/(X+1)$ is a domain, then so is $\mathbb Z[X/(X^2-1)$. What is the mistakes here ? Is there a way to have two copies of  app in Ubuntu each with a different account?  It seems copying the app in another folder does not work. Is there a number whose absolute value is negative? Is it possible to send an email from my Java application using a GMail account?  I have configured my company mail server with Java app to send email, but that's not going to cut it when I distribute the application. Answers with any of using Hotmail, Yahoo or GMail are acceptable. Why is my membership length only "Six months"? I was viewing the  and I noticed that I've apparently only had my account for six months, which is news to me... I opened my first site account here in February of 2014.  My account is much older than this, what's causing this bug?  To add some detail... On the per-site list, it does line up the &quot;member for X&quot; text with my actual membership time on that site. For example, when I look at the Meta Stack Exchange leaderboard, it shows me as a member for 2 years, 10 months. This is expected.  Update. With some work, today, I've managed to exceed my Interpersonal Skills hat count of with my Meta Stack Exchange hat count. This has changed my account age to match my MSE account age...  Still not reflective of my actual age... but better. Does causation imply correlation? Electrostatic charge leakage What are the ways electrostatic charged objects leak charge in humid conditions?  Can airborne particles pick up charge by contact, then be repelled hence removing charge? If so would it be a significant factor? Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things.