I am currently studying electrodynamics with all the fields and the like. Now, as I understand it, in a more modern viewpoint there is a duality between electromagnetic fields and photons, with photons being the particles that are exchanged in the process of interaction.  My question is, what is the current explanation to what an electromagnetic field is? For example, consider a point charge $q_1$. In order for another charge $q_2$ to 'feel' $q_1$, there is an electromagnetic field generated by $q_1$ that allows interaction. However, using the photon picture of view, a charge should then constantly radiate photons in all possible directions to let other charges know that it's here and should be interacted with. This leads to a problem in energy conservation, as each photon carries an energy $h\nu$, and thus even if a charge is at rest it would radiate off its energy and subsequently be gone.  How can this be resolved? What is the real connection between an electromagnetic field and photons?
I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons.  So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again!   But if it’s all virtual photons, how do we get the difference between a magnetic and electric field?  [This question needs the tag "magnetostatics" as well].
Are fractions with prime numerator and denominator dense? Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$? I want to decide whether or not $f_n(x)=nx^n(1-x)$ is uniformly convergent or not. I have shown that $\lim_{n\to\infty} f_n(x)=0$ for $x \in [0,1]$.  Now $f_n(0)=f_n(1)=0$. And in $(0,1)$, we have $f'_n(x)=n^2x^{n-1}-n(n+1)x^n$. Putting this to zero gives $x=\frac{n}{n+1}$. Also $f_n(\frac{n}{n+1})=(\frac{n}{n+1})^{n+1}$. But that tends to $\frac{1}{e}$ as $n$ tends to $\infty$ which is not $0$ so we don't have uniform convergence, right?  Nonetheless we have $\lim_{n\to\infty} \int_0^1 f_n(x)dx=\lim_{n\to\infty} (\frac{n}{n+2}-\frac{n}{n+1})=0=\int_0^1 \lim_{n\to\infty} f_n(x)dx$.  Is all this right? I have some doubts, I mostly doubt my intuition in analysis... Thank you! Penguin Brainteaser : 321-avoiding permutations There are $k$ penguins, $k\ge 3$. They are all different heights. How many ways are there to order the penguins in a line, left to right, so that we cannot find any three that are arranged tallest to shortest (in left to right order)? The penguin triples do not have have to be adjacent.  This problem is courtesy of someone I know only as "DaMancha." What is the best way to clean the sensor on a interchangeable lens camera (mirrorless or digital SLR)? JSON encode MySQL results How do I use the json_encode() function with MySQL query results? Do I need to iterate through the rows or can I just apply it to the entire results object? Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you. People like myself are Stack Exchange trusted users with thousands of reputation points and hundreds of badges.  In other words, we've proven ourselves as people who are assets to Stack Exchange and are a key component in making it so profitable.  As such, I was surprised when I posted  on the  site, and then shortly afterwards tried to  on that same meta site, but was blocked with a message in red stating I had to wait 40 minutes before posting again.  Why is this limit present for trusted Stack Exchange contributors, and can it be changed? Is this NPN transistor AND logic gate practical? Is it true that if $A$ is a nonnegative primitive matrix, then $AA^T$ is also primitive? Obviously $A^T$ is primitive but in general product of primitive matrices is not primitive.  Any hint? How To Make A Solid Object Drop To The Ground Like It Was Liquid So I have a simple bouncing ball animation, but I just wanted to something "weird" for the ending. Is there a way to change the ball's material on the fly, where after the last bounce it changes materials and physical composition from a rigid body(baked animation now) to a fluid sim?  (within 30-40 frames) Name an Armor Stand using the Standard Galactic Alphabet The Standard Galactic Alphabet is what appears in the Enchanting Table. Is there a way to give a name to an Armor Stand using these character.  I know there is a LingoJam Translator that give ASCII characters that look like the SGA, but that is not what I'm looking for. Can I use the same characters as the Enchanting Table? Determine the exact date of creation of an Apple ID Is it possible to determine the exact date and time (if possible) of creation of an Apple ID?   Is there any official or unofficial way to establish this if you don't have the initial sign up information recorded? How to Query all opportunities ids from account I write a sample query to fetch account Ids from cases of a particular record type.Then i wrote another sample query parent to child relationship to fetch opportunity ids from accounts.  My sample code  List&lt;Case&gt; c=new List&lt;Case&gt;(); c=[Select Id,accountId from case where RecordTypeId = '01290000000sF3L']; List&lt;Id&gt; accountIds= new List&lt;Id&gt;(); List&lt;Account&gt; accounts=new List&lt;Account&gt;(); List&lt;Opportunity&gt; opps=new List&lt;Opportunity&gt;(); for(Case cl:c)  { accountIds.add(cl.accountId); } accounts=[Select Id,name,(select Id,name from opportunities) from account        where Id in:accountIds]; System.debug('Data in accounts '+accounts);   But my issue is how to get opportunity id from above Soql accounts means how to add those opportunity id's to the list.  for(Account ao:accounts) { //How to fetch Opportunity ids from above account list. } Do photons bend spacetime or not? I have read this question:     where Safesphere says in a comment:     Actually, photons themselves don't bend spacetime. Intuitively, this is because photons can't emit gravitons, because, as any massless particles not experiencing time, photons can't decay by emitting anything. The latest theoretical results show that the gravitational field of a photon is not static, but a gravitational wave emanating from the events of the emission and absorption of the photon. Thus the spacetime is bent by the charged particles emitting or absorbing photons, but not by the photons themselves.        where John Rennie says:     as far as I know there has been no experimental evidence that light curves spacetime. We know that if GR is correct it must do, and all the experiments we've done have (so far) confirmed the predictions made by GR, so it seems very likely that light does indeed curve spacetime.   Now this cannot be right. One of them says photons do bend spacetime, since they do have stress-energy, but it is hard to measure it since the energy they carry is little compared to astronomical body's stress-energy. So they do bend spacetime, it is just that it is hard to measure it with our currently available devices.  Now the other one says that photons do not bend spacetime at all. It is only the emitting charge (fermion) that bends spacetime.  Which one is right? Do photons bend spacetime themselves because they do have stress-energy or do they not? rent a car, rent out a car, hire a car English not my native language. I'm trying to understand what would be correct in British English.  If I want to use a car for some time. I can say, "I want to rent a car". But from my knowledge there may be a  misunderstanding. I can say the same words if I want to give a car to someone to use for some time and I want to receive payment.  So what would be correct and not misunderstanding?   1) I want to use a car for some time = "I want to hire a car" 2) I want to give a car to someone to use for some time = "rent out a car"   Is my understanding correct? Why is     $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$   Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is     $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$     This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: . Splitting text files based on a regular expression I have a text file which I want to split into 64 unequal parts, according to the 64 hexagrams of the Yi Jing. Since the passage for each hexagram begins with some digit(s), a period, and two newlines, the regex should be pretty easy to write.  But how do I actually split the text file into 64 new files according to this regex? It seems like more of a task for perl. But maybe there's a more obvious way that I'm just totally missing. Does the alarm work when my phone is turned off? Will the alarm still go off if I turn off the phone before?  Example: when I have low battery, I have to go to sleep and wake up at X o'clock, and there is no charger around. Should I turn off the phone?   Note: If it makes any difference my phone is Galaxy S running froyo Address is invalid on local machine [on windows 8 only] What type of differential equation is $f'(x) = f(x/2)$ and how do you solve it? I have the following different equation $$f'(x) = f(x/2)$$ with $f(0)=10$. What type of DE is this, and how would you solve it? It seems $f(x)$ is likely to be some relative of $e^x$, since $f'(x) = f(x)$, which is close, but I don't even know what that type of DE is called with that $f(x/2)$ feature, so I'm not having any luck searching for a tutorial. The best candidate I've found was a &quot;delay differential equation&quot;, but that seems more suited to $f(x-3)$ than $f(x/2)$.