So my question is: why alternating current produces electromagnetic waves?   There are plenty of answers on google that state it as fact, but none answer why. I  mean what is it about accelerating electrons that generate EM wave? Are the theories for it? Is there lazy explanation?
Let's consider it case by case:  Case 1: Charged particle is at rest. It has an electric field around it. No problem. That is its property.  Case 2: Charged particle started moving (it's accelerating). We were told that it starts radiating EM radiation. Why? What happened to it? What made it do this?  Follow up question: Suppose a charged particle is placed in uniform electric field. It accelerates because of electric force it experiences. Then work done by the electric field should not be equal to change in its kinetic energy, right? It should be equal to change in K.E + energy it has radiated in the form of EM Waves. But then, why don't we take energy radiated into consideration when solving problems? (I'm tutoring grade 12 students. I never encountered a problem in which energy radiated is considered.)
Proving Invertibility and Eigenvalues If matrix $A$ is an $n\times n$ matrix such that $A^2 -A -2I=0$.  How can I show that $A$ is invertible and that $A^{-1} = \frac12(A-I)$? Also, how do I show that one of the eigenvalues of $A$ is $2$ or $ -1$? I appreciate any help on this question. How to replace spaces with newlines/enter in a text-file? I have simple text file named "example".  Reading with terminal command: cat example  Output:  abc cdef ghi jk lmnopq rst uv wxyz   I want to convert (transform) into following form: (expected output from cat example)  abc cdef ghi jk lmnopq rst uv wxyz   How can I do this via the command-line?   (This is only an example file, I want to convert word's position in vertical-column) Running Commands on Remote Linux Server Using SSH I am writing a Bash Script wherein I need to run some set of commands on a remote Linux server from my local computer using SSH. I want the results to be displayed on the terminal screen or dump the results in a file on my local system.   How can achieve this? Is the following syntax correct?  \#!/bin/bash . . . .  ssh &lt;user&gt;@&lt;remote_host&gt; 'COMMAND &gt;&gt; /path/to/file ; scp /path/to/file &lt;user&gt;@&lt;local_host&gt;:&lt;location&gt;; exit' . .   Thanks in advance. Global keyboard capture in C# application I want to capture a keyboard shortcut in my application and trigger a dialog to appear if the user presses a keyboard combo even outside of the app. Similar to Google Desktop Search's Ctrl, Ctrl to bring up the search dialog.  I have tried using some keyboard hook modules out there that basically use Win32 interop to get this effect but each implementation I've tried ties down the keyboard to some extent to where you start getting weird behaviors when the application is doing something intensive. Such as loading a large amount of data, this would cause the keyboard and mouse to lockup.  I'm looking for a lightweight solution that would allow this to be done without tying down the keyboard and mouse. "For security reasons, framing is not allowed; click OK to remove the frames." on Imgur upload Ok first, there's that lugubrious use of the semicolon; not necessary.  Second, I didn't open a frame!  You did!  Alls I tried to do is upload a 5MB animated .gif file via Imgur.    Looking further into this, Imgur actually posts an error when you upload an animated .gif larger than 2MB.    Go figure! If $A = \tan6^{\circ} \tan42^{\circ},~~B = \cot 66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$ My trigonometric problem is:     If $A = \tan6^{\circ} \tan42^{\circ}$ B = cot$66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$.    Working :   $$B = \cot 66^{\circ} \cot78^{\circ} = 1- \frac{\tan24^{\circ}+\tan18^{\circ}}{\tan42^{\circ}}$$  $$A= \tan6^{\circ} \tan42^{\circ} = 1- \frac{\tan6^{\circ} +\tan42^{\circ}}{\tan48^{\circ}}$$  but it seems  this is the wrong way of doing this...please suggest. Thanks! Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A In the next proof we avail ourselves of the next lemma: For all sets A and B, ( A ∪ B ) - B = A. Proof: Let A and B be arbitrary sets and let x ∈ ( A ∪ B ) - B. x ∈ ( A ∪ B ) - B ⇔ x ∈ ( A ∪ B ) ∧ x ∉ B ⇔ ( x ∈ A ∨ x ∈ B ) ∧ x ∉ B ⇔ ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B) For p ∧ ~ p ≡ F and p ∧ q ⇒ p where p and q are prepositions, ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B ) ⇔ x ∈ A ∨ F By modus tollendo ponens, x ∈ A ∨ F ⇔ x ∈ A So then, x ∈ ( A ∪ B ) - B ⇔ x ∈ A And ( A ∪ B ) - B = A Therefore, the statement for all sets A and B, A ∩ B = ∅  implies ( A ∪ B ) - B = A is true trivially. Is this proof right? The topic which haunted me for two years until I gave up on it. But now I am doing engineering and this topic suddenly popped out of my textbook from nowhere. I seriously need to understand this topic because I wasn't able to do so in the past. I have read many book on 'energy' and it came to nothing. Maybe just because the many books I have read all said 'energy' is something which doesn't 'exist'! It is 'abstract'! It is just a number which represents the state or orientation of a system. But then I see so many examples that 'uses' energy to do 'work'. But the question here is that if anything doesn't exist in this universe, then how can the same thing be used to do something that exist?  My problem is that I didn't understood the topic 'energy' and all the other topics related to it (work, power etc.) The normal JOIN ... ON ... syntax is well known. But it is also possible to position the ON clause separately from the JOIN that it corresponds to. This is something that is rarely seen in practice, not found in tutorials and I have not found any web resource that even mentions that this is possible.  Here is a script to play around with:  SELECT * INTO #widgets1 FROM (VALUES (1), (2), (3)) x(WidgetID)   SELECT * INTO #widgets2 FROM (VALUES (1, 'SomeValue1'), (2, 'SomeValue2'), (3, 'SomeValue3')) x(WidgetID, SomeValue)  SELECT * INTO #widgetProperties FROM (VALUES     (1, 'a'), (1, 'b'),     (2, 'a'), (2, 'b')) x(WidgetID, PropertyName)   --q1 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN #widgets2 w2 ON w2.WidgetID = w1.WidgetID LEFT JOIN #widgetProperties wp ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b' ORDER BY w1.WidgetID   --q2 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN #widgets2 w2 --no ON clause here JOIN #widgetProperties wp  ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b'  ON w2.WidgetID = w1.WidgetID ORDER BY w1.WidgetID   --q3 SELECT w1.WidgetID, w2.SomeValue, wp.PropertyName FROM #widgets1 w1 LEFT JOIN (     #widgets2 w2 --no SELECT or FROM here     JOIN #widgetProperties wp     ON w2.WidgetID = wp.WidgetID AND wp.PropertyName = 'b') ON w2.WidgetID = w1.WidgetID ORDER BY w1.WidgetID   q1 looks normal. q2 and q3 have these unusual positionings of the ON clause.  This script does not necessarily make much sense. It was hard for me to contrive a meaningful scenario.  So what do these unusual syntax patterns mean? How is this defined? I noticed that not all positions and orderings for the two ON clauses are allowed. What are the rules governing this?  Also is it ever a good idea to write queries like this? What does "can only heal so much" mean in this sentence? Visit to US before F1 using the Visa Waiver Program I am going to start a phd at the US with an F1 visa. University told me that I could enter the US only 30 days before the date on I-20. Since English is not my first language (and this would be a big change in my life) I would like to go there by June.  The thing is that I have an European passport with a valid ESTA. My plan would be to arrive with this passport, stay 4-6 weeks , go to Canada for a short vacation (maybe flight maybe by land) and come back to the USA with F1 visa.  I have two questions about this:  1) is this possible?   2) If yes, how could I justify knowing I will lose my come-back flight during my first entry? (I would purchase round trip because one-way are too expensive, probably with less than 90 days difference) I plugged in my external harddrive (which was formatted on my Mac into HFS+ journaled) to my Ubuntu desktop 9.04 64bit.  I am not able to get the drive to mount with write capability, how do I do that?  Right now all I'm getting is read access, I tried  sudo mount -t hfsplus /dev/sdf2 /media/"Portable HD"   but that still gave me only read access...  ideas?? Vector spaces for the irreducible representations of the Lorentz Group EDIT:  The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain unanswered.  The definition of a representation is a map (a homomorphism) to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the   $(0,0)$ Representation $(\frac{1}{2},0)$ Representation  $(0,\frac{1}{2})$ Representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ Representation $(\frac{1}{2},\frac{1}{2})$ Representation infinite dimensional Representation? Is the sign of principal components meaningless? Explain the xkcd jelly bean comic: What makes it funny? How to upgrade Ubuntu from an ISO image I'm using Ubuntu 10.04 now and want to Upgrade to Ubuntu 11.04. The problem is my net connection is very slow and I can not do a network upgrade from the "Updates Manager".  So, is there a way that I can download the Ubuntu upgrade as an bootable CD image and upgrade from it? Types in MySQL: BigInt(20) vs Int(20) Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric $\sum_{k=1}^nH_k = (n+1)H_n-n$. Why? Is this asbestos? (South korea) School in South Korea removed asbestos over the past few weeks. Now I am all paranoid.   Is this asbestos? Found these dime-sized chunks by my desk.          Higher resolution images are available here:    There should be 4 pics. If not clear enough let me know.