A spherical shell, uniformly charged, exerts no field inside it ; the point being either at centre or off-centre. According to Gauss law if we construct a spherical Gaussian surface all inside the shell, it contains no charge inside and thereby zero net flux on that surface. Now, here we usually conclude that the field is zero as flux is zero. My question is how can we say the field is zero,  whereas the flux can also be zero in other configurations, where ingoing and outgoing field lines are equal in number. For instance, spherically symmetrical $1/r^2$ field and the source point being outside Gaussian surface.
If you have a conducting hollow sphere with a uniform charge on its surface, then will the electric field at every point inside the shell be 0. The reason the electric field is 0 at the center is clear from the symmetry of the sphere, but for a point at a certain distance from the center, shouldn't a net electric field exist?  I saw a similar question on the site but it was vaguely explained.
How to do vlookup and fill down (like in Excel) in R? Finding the solution of the congruence $4x\equiv16\pmod{26}$ Solve the congruence   $$4x\equiv16\mod{26}.$$  How do I find the solution to this? I have tried by the euclidean algorithm but the gcd is not $1$ so it doesn't work.  $$\begin{align} 26&amp;=&amp;6&amp;\times4&amp;+2\\ 4&amp;=&amp;2&amp;\times2&amp;+0 \end{align}$$  Note: I understand we can see that $4$ and $17$ are solutions but how would I work this out algorithmically? Does an absolutely integrable function tend to $0$ as its argument tends to infinity? Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that   $$\int_{0}^\infty|f(t)|dt&lt;\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$  If so can you provide a proof, otherwise a counter example. Thank you. How do I enter a strong (long) DKIM key into DNS? Transferring a Windows 8 license and proper un- and reinstallation Long story short  I have two computers: a laptop and a desktop computer. Both have Windows 7 on them.  I buy the Windows 8 Pro upgrade. To see if it screws up anything, I install it on my laptop as a guinea pig. I intend to use Windows 8 for my main computer, my desktop, but I want to test it on the laptop, so I know I don’t risk losing access to my desktop and the data on it. I never use my laptop, and only used it, because it already has a Windows 7 installation on it.  The problem  At some point, I must have entered the license key on my laptop, because when I go to the activation screen on my desktop, I get this:    Uh-oh. I can’t use the key on my desktop. Now how the hell do I transfer the key from my laptop to my desktop computer?  Answers and suggestions so far  Let’s just say that I tried everything possible to get some answers on this matter. The best response I got from Microsoft is this:     To install Windows 8 on your desktop, do the following:         Uninstall Windows 8 on your laptop   Afterwards, install Windows 8 on your desktop   If it won’t activate, call product activation at (...)      I am not a fan of that last point. The error message does allude to such a solution, however:     If you’ve reinstalled Windows or made changes to your hardware recently, you may be able to use your current key.   The question  My main question is this: has anyone been in a similar situation, and if so, what did you do to resolve this?  Failing that, what is the proper way to   uninstall the Windows 8 installation on my laptop, and reinstall the Windows 8 installation on my desktop?   Ad 1  I have already tried using the “reset” feature on my laptop, but that only resulted in a new Windows 8 installation that was already activated. But which is the right way to uninstall the installation in a way that allows me to use the license key on the desktop computer?  Ad 2  Which is the proper way to reinstall the Windows 8 installation on my desktop computer? Why do I even have to reinstall it in the first place?  I won’t get around to do this, until my USB key with 3.0 support arrives in the mail, but it is going to be a while, until I find a assuaging response to the best way to go about this anyway. Default class inheritance access How to delete all music on my iPhone "is" vs "are" when followed by a number I read the following sentence on YouTube and initially thought it was grammatically incorrect:     Here are 10 minutes of the movie Black Hawk Down!   I thought "are" should have been "is" but then I realized minutes is plural so the plural form, are, is used correctly. I also thought how most people would naturally say "Here is ten dollars in change" but is the use of the singular form, is, correct since dollars is plural?   Am I misunderstanding a rule? Mismatch Reputation in the thumbnail and the actual Reputation Actually this was quite surprising. A casual troll in the Users Page revealed few users with negative vote. One example is here    Then trying to see what could have causes such a low reputation, I peeked inside the Profile Page of this particular User and was surprised to see a decent positive score.    Can someone explain to me how this can be possible? How do you convert a number to a string showing dollars and cents?  eg: 123.45    =&gt; '$123.45' 123.456   =&gt; '$123.46' 123       =&gt; '$123.00' .13       =&gt; '$0.13' .1        =&gt; '$0.10' 0         =&gt; '$0.00' Is black a color or absence of color? What are exchanges like "How are you," "I'm fine," and "See you later" called? How to do a secure erase on a Macbook Pro? I have a Macbook Pro with SSD and I would like to completely erase every single bits of data on it. How would I do that? On a PC, I would just run Darik's Boot And Nuke, but can I run it on a Mac too, or is there a better alternative? exponentiation of cardinal number How do I change file permissions on a FAT32 drive? Don't let the association bonus be given away as bounties When a user with at least 200 rep on any Stack Exchange site registers a new account on another SE site, they immediately receive 100 rep on that site as an .  This is a deliberate feature to give users who are already familiar with how the SE platform and community works a bit of an extra boost (e.g. letting them upvote and comment immediately), and it's pretty much working as intended. What's probably not intended is that, currently, this bonus rep can be given away as . In fact, because awarding bounties currently requires only 75 rep, any user with 200+ rep on at least one site can join another site, receive the 100 rep bonus, and immediately award it all as a bounty to another user, without having to earn any actual rep on the site at all.  Not only does this seem contrary to the design intent of bounties (and of the association bonus), but it also seems ripe for potential abuse. (In fact, the only thing stopping a sockpuppet ring from gaining unlimited rep this way is the fact that the bounties would likely be sooner or later noticed by mods.) It's also possible for experienced users to (more or less) accidentally bounty their reputation so low that that they lose access to basic site features that they'd normally have due to the bonus.  Arguably, of course, this is entirely the user's own fault, but it can still be .  Also, this behavior , which currently require 110 rep to answer for users with the association bonus. The fix I'd like to suggest is simple:  If a user has been awarded the association bonus, don't let them award a bounty that would take their rep below 100.  Of course, it would still be possible for users with the bonus to drop below 100 rep via other means, e.g. by having their posts downvoted.  But they wouldn't be able to just give away this baseline rep to other users any more. Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46. I'm a US passport holder. What documents do I need when I fly from the U.S. to Germany?  Assume I'm going for a few weeks of vacation.  Obviously I need my passport.  What about a "visa"?  Anything else? Prove that a function is not injective Should I let potential employers know I have a job offer? I've interviewed for a few positions, one of which has made me an offer.  I'm very pleased with the offer, but I prefer some of the other positions.  I have the luxury of being able to take a lot of time to decide whether or not I accept this offer, so I'm certain I will hear back (either positively or negatively) from the other positions before a decision is required for the offer I've received.  I'm trying to decide between two courses of action, and I want to choose the one that will best help my candidacy with the positions I've yet to hear from.  Plan 1 is to do nothing and simply wait to hear back from the other positions for which I've already interviewed.  Plan 2 is to be in touch with the other positions to let them know I have an offer, and ask where I stand with them.  Would Plan 2 make me a more desirable candidate with the other positions, as they might see my offer as further validation of my credentials?  Or, might initiating with the interviewers in this way be detrimental to my candidacy, perhaps making me seem pushy or self-centered?  If so, I would rather just wait for them to run their course of action (Plan 1).  As I've said, I have plenty of time, so I don't want impatience to hurt me.  If there is a good chance that my meddling will be detrimental, then I'd rather be hands-off here.  However, if it's more likely that informing the other interviewers of my offer will give me a boost, then I'd want to do so.  Thanks for any clarity you can provide!  Edit: For a bit more information and context, the position I've been offered is a non-academic research position that my advisor has described as "prestigious."  The other positions for which I've interviewed (and which I prefer) are mostly academic teaching-centered positions (visiting assistant professors, tenure-track at smaller four year institutions).