How does the combination of different medium affect the force? How do we calculate the equivalent dielectric constant? Say, two charges are separated by 100 cm, first 10 cm is a slab of dielectric constant 100, the last 10 cm is another slab of constant 50, rest is air.
I am given two charged particles of same charge at a distance of $r$. They initially apply force $F$.  Now an infinite dielectric (of dielectric constant $4$) of width $\frac{r}{2}$ is introduced between the particles. What will be the new force?   I find this problem confusing because I have only been told about forces when its either fully dielectric or fully vaccum given by Coulomb's law. How do we get forces when only partial space is dielectric?
I know I need to have (although I don't know why) a GROUP BY clause on the end of a SQL query that uses any aggregate functions like count, sum, avg, etc:  SELECT count(userID), userName FROM users GROUP BY userName   When else would GROUP BY be useful, and what are the performance ramifications? How do I deploy an OpenStack cloud with Juju? I'd like to get a test cloud up and running in preparation for a possible deployment of 12.04. I messed with the UEC a bit, but have had a much harder time finding anything helpful as a primer for the new stack. There is generic documentation on Diablo, but there are likely to be a lot of pieces already handy in Ubuntu that a specific guide could save time with.  Also, as much as I would love to, my budget doesn't include the $4-6k to get a small cluster running with the Canonical support. How to prove this decomposition of sum of squares? I am reading a paper   It has a mathematical step shown as  $$\sum_{i=1}^{n}(x_i-\mu)^2=n(\mu-\overline x)+\sum_{i=1}^n(x_i-\overline x)^2$$  Here the $\mu$ is unknown parameter and $\overline x$ is the known mean of $\mathbf x=(x_1, x_2, x_3, \dots,x_n)$  How do I demonstrate this identity? Unable to set DYLD_FALLBACK_LIBRARY_PATH in shell on OSX 10.11.1 In shell scripts used for unit testing with dynamic libraries in a directory other than the typical @rpath, I have previously been able to set DYLD_FALLBACK_LIBRARY_PATH to set the directory containing the libraries. Under 10.11.1, bash seems to ignore attempts to set this environment variable:  $ sh -x testscript.sh + DYLD_FALLBACK_LIBRARY_PATH=/Users/something/testinglibs + export DYLD_FALLBACK_LIBRARY_PATH + exec printenv   and DYLD_FALLBACK_LIBRARY_PATH is not present in printenv's output.  Is this a security-related hack in 10.11's shell? I haven't been able to find this change documented in man pages or online. Open-source alternative to Adobe InDesign There is some open-source software, similar to Adobe InDesign? It would be awesome if it do exist...  I know all the others like Inkscape, Gimp, etc... but for InDesign i have none.  Could you give me one direction? migrating / merging iphone contacts I have an iPhone 6 (with proper iTunes backup) where all of my contacts where, but I bought a new iPhone 7 and had to initialize it as new iPhone, due to several time-related constraints, so now I have a few but new contacts on my new phone, but none of the older ones, but would be the best practice oriented, for just recovering the old contacts from the iTunes backups?  PS: I still have the old iPhone 6, so it could be used as "temporary storage" if needed. Why is ++i considered an l-value, but i++ is not? Why is ++i is l-value and i++ not? Let $m$ and $n$ be two integers. Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.   I think I have to use the contrapositive to solve this. So I assume $\neg P\implies Q$ and I have to derive $\neg Q$? I know its easier to show odd numbers rather than even... but im still unsure how to do this practice problem. How to make putty to not break my session after some time? Each time putty is closing the session after some time if it is idle.  There is no time parameter on putty, so how can I keep my putty ssh session always Alive? What is accountsd and why does it want to use the "login" keychain? How do I drop root privileges in shell scripts? The "--up" option in OpenVPN is normally used for routing etc. And so it is processed before OpenVPN drops root privileges to run as nobody. However, I am invoking shell scripts that need to run as an unprivileged user.  How do I do that? I have studied , especially polynomial's and tylerl's answers, but I do not understand how to implement. I am working in Centos 6.5, and suid is blocked, both as "chmod u+s" and as "setuid()".  There is an OpenVPN plugin ("openvpn-down-root.so") which enables scripts invoked by the "--down" option to run as root. There could be an equivalent, such as "openvpn-up-user.so", but I have not found it.  Edit0  Per Nikola Kotur's answer, I've installed . Although the chpst command works in terminal, in the up script it fails with "command not found". What works is "sudo chpst" plus setting the proper display and language. Please see  Given that, the up script needs these four lines:  LANG="en_US.UTF-8"; export LANG GDM_LANG="en_US.UTF-8"; export GDM_LANG DISPLAY=:0; export DISPLAY sudo chpst -u user -U user /home/user/unprivileged.sh &amp;   Edit1  Per 0xC0000022L's comment, I find that "sudo -u user" works as well as "sudo chpst -u user -U user":  LANG="en_US.UTF-8"; export LANG GDM_LANG="en_US.UTF-8"; export GDM_LANG DISPLAY=:0; export DISPLAY sudo -u user /home/user/unprivileged.sh &amp;   I'll study man sudoers and update if/when I get sudo alone to work. What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$? Omission of 'for' with various quantified time intervals: influence of verb Device not detected in Eclipse when connected with USB cable We now that Fermat's last theorem is true so there are not positive integer solutions to $$x^n+y^n=z^n$$ for $n\in\mathbb{N}$ and $n&gt;2$.  But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$? How can I display site logo, name and slogan? How to UPSERT (MERGE, INSERT ... ON DUPLICATE UPDATE) in PostgreSQL? How to prove $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$? there is a similar thread here , but I'm only looking for a simple linear algebra proof. How can I determine the current brew package count? I tried to get the number of available brew packages by running  brew search | wc -l   but I don't know how accurate this really is. What do I need to take into account to get a more correct number?  PS: macports lists 19,889 packages online Why is HMAC-SHA1 still considered secure? This Q &amp; A   says that the security of HMAC-SHA1 does not depend on resistance to collisions? Are they are saying specifically with respect to HOTP or for HMAC-SHA1 for any use?     If HMAC-SHA1 is used for verifying the integrity of a document and you can find a collision, you can make changes in the document, right?      Even the RFC says that resistance to collisions is relevant       The security of the message authentication mechanism presented here depends on cryptographic properties of the hash function H: the resistance to collision finding (limited to the case where the initial value is secret and random, and where the output of the function is not explicitly available to the attacker), and the message authentication property of the compression function of H when applied to single blocks    What am I missing here?