Can someone provide a derivation of the  for electromagnetic induction? To be clear,  $$ d\vec{B}~=~\frac{\mu_0}{4\pi}\frac{I d\vec{\ell}\times \vec{r}}{r^3}. $$  Is there a simple way to compute the magnetic field at a point between two Helmholtz coils, if the radii of the coils are the same and the current through each coil is the same?
There's already a question like this  so that my question could be considered duplicate, but I'll try to make my point clear that this is a different question.   Is there a way to derive  from the Lorentz' Force law or just from Maxwell's Equations?   The point is that we usually define, based on experiments, that the force felt by a moving charge on the presence of a magnetic field is $\mathbf {F} = q\mathbf{v}\times \mathbf{B}$, but in that case the magnetic field is usually left to be defined later.   Now can that force law be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law?  I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument.   In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?
It is will known that increasing cooling of a metal casting will lead to finer micro-structure which mean higher mechanical properties. If I have 2 molds one is thick copper die ,&amp; second is just thin-walled copper cup or mold cooled by water. which one of them excepted to cool the molten metal i.e. aluminium faster ?   by calculation it seems that copper die have higher cooling power as it has higher thermal diffusivity , but in real life it seems water have more cooling power. I am so confused. How can I really disable the Cmd to Win key mapping in Parallels Desktop 5? Removed it from System Preferences &gt; Keyboard. No go. Remapped ⌘ Cmd to be Ctrl. No go. Ripping out the ⌘ Cmd key shouldn't be the solution. Anyone?... If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?  I think the $Ker\phi = \{x\in G:\phi(x)=1\}$  because you are looking at an identity then it is all reals $0$ does not work a because if $x=0, e'x=0$  but then i realize it is injective how does that affect the kernel? Access to journals for individuals outside of an institution? For all that is said about the wealth of information that is freely available on the internet, the fact is that most of it is incomplete, dumbed down, lacking in context or downright wrong.   Individuals spend the best part of 20 years completely oblivious to the existence of journals and the process of peer review and even then we'd be lucky if 10% of those with access to journals (usually through an institution) actually bother to invest in them.   Granted that some journals will be totally inaccessible to those without a solid background in some particular field but by and large exposure is a good thing and there are many areas of research where individuals will benefit directly from having read these.  Where does this come from?  I decided I wanted to read Popular Politics in the Late Medieval City: York and Bruge in The English Historical Review today, mainly because it looks interesting and I'm hoping to invest more time into learning about British and European History. Incidentally I also wouldn't mind tucking into Super Stable Clocks, Nature 500, 505 (29 August 2013) and a number of data analysis and big data journals in order to advance my career.   Now, not being affiliated with an institution that subscribes to these journals means that I would typically have to pay between £5 ($10) and £15 ($25) per article that I read... I could probably spend £100 just to pass an hour by.  Question   What's being done to address the lockdown on 'higher' education material? I've noticed that the Directory of Open Access Journals publishes open articles and other firms are doing similar, mainly with journals in new fields? How could I, as an individual outside of an institution, gain journal access without it costing so much money? Fullscreen YouTube video crash "You just can't" vs. "you can't just " I'm a bit confused about this. Which expression is correct?     You can't just do that.    or      You just can't do that.   I'm trying to say:     You can't just bash an ideology because of what someone has said/done. How to build a dmg Mac OS X file (on a non-Mac platform)? Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? Installing 3rd party python libraries for QGIS on Windows "He was telling me that he is going..." vs. "He told me that he is going..." Looping through array and removing items, without breaking for loop How can I share a directory with an another user? I don't want the shared folder to be accessible by anyone on the system, I want it to be accessible just to a restricted set of users.  How to I do that? I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands! What are 'closures' in .NET? What is a closure? Do we have them in .NET?  If they do exist in .NET, could you please provide a code snippet (preferably in C#) explaining it? I've got an HP Pavilion Touchsmart notebook that came with Windows 8. I've upgraded to 8.1, and now I want to dual-boot an Ubuntu distribution with Windows 10. When dual-booting with Windows 8.1, the Windows Boot Manager always took priority, even after installing GRUB. What do I need to do to get GRUB to be the default bootloader instead of Windows Boot Manager? window.onbeforeunload and window.onunload is not working in Firefox, Safari, Opera? How to capture the scale of a scene? Can I use a present-tense verb when writing about a past event? char!=(signed char), char!=(unsigned char) The code below compiles, but has different behavior for the char type than for the int types.  In particular     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;   result in 3 instantiations of templates for three types: int8, uint8 and char. What gives?  The same is not true for ints: int and uint32 which result in the same template instantiation, and signed int another.  The reason seems to be that C++ sees char, signed char and unsigned char as three different types. Whereas int is the same as a signed int. Is this right or am I missing something?  #include &lt;iostream&gt;  using namespace std;  typedef   signed char       int8; typedef unsigned char      uint8; typedef   signed short      int16; typedef unsigned short     uint16; typedef   signed int        int32; typedef unsigned int       uint32; typedef   signed long long  int64; typedef unsigned long long uint64;  struct TrueType {}; struct FalseType {};  template &lt;typename T&gt; struct isX {    typedef typename T::ikIsX ikIsX; };   // This  int==int32 is ambiguous //template &lt;&gt;            struct isX&lt;int  &gt;    { typedef FalseType ikIsX; };  // Fails template &lt;&gt;            struct isX&lt;int32  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint32 &gt;  { typedef FalseType ikIsX; };   // Whay isn't this ambiguous? char==int8 template &lt;&gt;            struct isX&lt;char  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;int8  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint8 &gt;  { typedef FalseType ikIsX; };   template &lt;typename T&gt; bool getIsTrue(); template &lt;&gt;           bool getIsTrue&lt;TrueType&gt;() { return true; } template &lt;&gt;           bool getIsTrue&lt;FalseType&gt;() { return false; }  int main(int, char **t ) {    cout &lt;&lt; sizeof(int8) &lt;&lt; endl;  // 1    cout &lt;&lt; sizeof(uint8) &lt;&lt; endl; // 1    cout &lt;&lt; sizeof(char) &lt;&lt; endl;  // 1     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;     cout &lt;&lt; getIsTrue&lt; isX&lt;int32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;int&gt;::ikIsX  &gt;() &lt;&lt; endl;  }   I'm using g++ 4.something Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? Similes and Metaphors - are similes a subset of metaphors?