This is a sum in a reference book( H.C. Verma, concepts of phycics). A dielectric slab is being inserted in a parallel plate capacitor as shown in the diagram. The width of the plates is given to be $b$. We need to find the force exerted by the capacitor on the slab. The solution given in the book is as follows: As voltage across the capacitor is same, the two sections are in parallel $\therefore c = c_1 + c_2 =\frac {\epsilon_0 b} {d} [l + x(k-1)]$ $ U=\frac 1 2 cV^2$ $F=-\frac {dU}{dx} = -\frac {V^2}{2} \frac {dc}{dx} = -\frac {\epsilon_0 b V^2(K-1)}{2d}$ I have some doubts about this solution. Firstly, the force is independant of $x$ which is a bit weird. We can see that the force is zero when $x=l$. Thus, the potential energy should have a maxima at $x=l$, which isn't seen here. Secondly, the charge on the plates is continuously changing, so current is flowing from the battery. Thus, there should be some heat that is generated, which is not taken into account.
In the book Principles of Physics by Resnick Halliday: The decrease in Potential energy of a parallel plate capacitor due to a dielectric is because the slab would start to oscillate and the energy would transfer back and forth between the kinetic energy of the moving slab and potential energy stored in the electric field. First of, can somebody provide me the math of the SHM and could somebody explain why it would become the potential of the capacitor i.e. the electric field and not the potential of the slab? As the slab is doing SHM, shouldn't the slab(and not the capacitor) have potential energy at one point of time and kinetic at another like a spring? If the SHM is because of the electric field due to the conducting plates in capacitors, won't they always be in one direction (Plus, it's magnitude is not even a function of distance!)? Could someone be kind enough to provide the math and the reason for Potential and Kinetic Energy transfer question?
How do electromagnetic waves wave? Electromagnetic waves have a physical crest and trough as observed in microwaves and radio waves. I understand that is electromagnetic field vectors that wave, not the photon. But how do they wave? What does causes them to change direction? Why does "unisex" mean both sexes? There are some places where men and women are segregated — for example public toilets and public swimming pool changing areas.  By this I mean, for example, that there are "Men's toilets" and "Ladies' toilets".  However, in some more cosmopolitan swimming baths, and other places, there are such things as unisex changing areas, or unisex toilets.  From my understanding uni- implies one, as in universe, unilateral or unicycle.  However when it comes to unisex, this means both sexes, so applies to two things.  How does uni- mean one in unisex? Does it mean one? Ubuntu Software doesn't open in Ubuntu 20.04 How can we run a command stored in a variable? $ ls -l /tmp/test/my\ dir/ total 0   I was wondering why the following ways to run the above command fail or succeed?  $ abc='ls -l "/tmp/test/my dir"'  $ $abc ls: cannot access '"/tmp/test/my': No such file or directory ls: cannot access 'dir"': No such file or directory  $ "$abc" bash: ls -l "/tmp/test/my dir": No such file or directory  $ bash -c $abc 'my dir'  $ bash -c "$abc" total 0  $ eval $abc total 0  $ eval "$abc" total 0 What is time dilation really? Please will someone explain what time dilation really is and how it occurs? There are lots of questions and answers going into how to calculate time dilation, but none that give an intuitive feel for how it happens. "Die from cancer" vs. "die of cancer" Is there a difference between those expressions: "Die from cancer" or "Die of cancer"? Are they both correct? Why use sucrose rather than glucose in an injectable vaccine? How do I specify unique constraint for multiple columns in MySQL? I have a table:  table votes (     id,     user,     email,     address,     primary key(id), );   Now I want to make the columns user, email, address unique (together).  How do I do this in MySql?  Of course the example is just... an example. So please don't worry about the semantics. Do quest items weigh anything? Someone said that quest items don't count against your carrying capacity in Skyrim, even if they have a weight listed. Is this false? How to access a SQL Server database from other computer connected to the same workgroup? I have created a C# application which uses a SQL Server database. I have other computers connected to me and to each other in a workgroup. I have shared my C# application with others. When they open the application they get the error      A network related or instance-specific error occured while establishing a connection to SQL Server. the server was not found or was not accessible   But the application is working fine on my PC. The connection string I am using is   Data Source=ASHISHPC1\SQLEXPRESS;Initial Catalog=ACW;User ID=ash159;Password=ashish159   which is stored in a .config file.   The application is working fine on my PC. What must I do? I have enabled the TCP/IP in the server but the same error persists. Some change in connection string or something else? Getting new files to inherit group permissions on Linux Why are passports checked at the check-in counter, Immigration and even at the gate? I was wondering, why there are passport checks at the check-in counter, Immigration and even at the gate?    For example, I was at Istanbul flying to either UK or US. There's a security check at the check-in counter asking a bunch of questions about my immigration status - even more than the Immigration officer at the border. Also, at the gate, there's also Security officer checking my immigration status and passport asking the same questions.    I thought that would be the responsibility of an Immigration officer rather than some third party company?   The thing I'm curious the most is they don't even know which countries this passport can go to with or without visa. They have to go back to the counter and check with the system which takes a long time. I was wondering why they hire these kind of companies to slow down the process? Algebraic Property inherited by a "lift"? Suppose $K$ is a field, and $K\subset E$ is an algebraic extension, and $K\subset F$ is any extension.  Suppose that $L$ is some field which contains both $F$ and $E$ as subfields.  I want to show that $FE$ (the smallest subfield of $L$ containing both $F$ and $E$) is algebraic over $F$.  Lang mentions that the result is true because an element remains algebraic under lifting, but I can't prove the claim with this.  I interpret his comment to mean that if $x\in E$ is algebraic over $K$, then $x\in FE$ is algebraic over $F$.  This is certainly true.    But this doesn't seem enough if I start with any element of $FE$. I am asked to fill the genotypes in the spaces provided but looks like I am aving a little bit of trouble. can anyone help? thanks in advance! Is there any way to increase the drop rate of items? "DataBinding: 'index+NewsItem' does not contain a property with the name 'Link'", but property exists (Not a Typo) multiple regression and multiple comparisons Say I fit a multiple regression of p explanatory variables. The t-test will allow me to check if any single one of those is significant ($H_0: \beta_i = 0$). I can do a partial F-test to check if some subset of them is significant ($H_0: \beta_i=\beta_j=...=\beta_k=0$).  What I often see though is that someone gets 5 p-values from 5 t-tests (assuming they had 5 covariates) and only keeps the ones with a p-value &lt; 0.05. That seems a bit incorrect as there really should be a multiple comparison check no? Is it really fair to say something like $\beta_1$ and $\beta_2$ are significant but $\beta_3$, $\beta_4$ and $\beta_5$ are not?   On a related note, say I run 2 regressions on 2 separate models (different outcome). Does there need to be a multiple comparison check for significant parameters between the two outcomes?  Edit: To differentiate from the similar question, is there any other interpretation to the p-values besides: "B_i is (in)significant, when adjusting for all the other covariates"? It doesn't seem that this interpretation allows me to look at every B_i and drop those less than 0.5 (which is similar to the other post).   It seems to me that a sure fire way to test whether B_i and Y have an a relationship would be to get a correlation coefficient p-value  for each covariate and then do a multcomp (although that would definitely lose signal).  Finally, say I computed the correlation between B1/Y1, B2/Y1 and B3/Y1  (thus three p-values). Unrelatedly, I also did a correlation between T1/Y2, T2/Y2, T3/Y2. I'm assuming the correct Bonferroni adjustment would be 6 for all 6 tests together (rather than 3 for the first group and 3 for the second group - and thus getting 2 "semi"-adjusted p-values). convergence of $\sum_{n=1}^{\infty }\frac{(-1)^{[\sqrt{n}]}}{n}$ where $[a]$ means floor function of $a$ I have a problem with solving the convergence of $\sum_{n=1}^{\infty }\frac{(-1)^{[\sqrt{n}]}}{n}$ where $[a]$ means floor function of $a$. I've already made the first thep and rewrote it with the $\sum_{n=1}^{\infty }(-1)^{n}\sum_{k=n^2}^{(n+1)^2-1}\frac{1}{k}$, which converges if and only if the previous one converges. Now I want to use Leibniz criterion, but I cannot prove that the second sum in the previous expression is monotonic. Does anyone know how to do it? (I cannot use Riemann integral, we haven't defined it yet) Let $X_1, \cdots, X_n$ be a random sample from the PDF: $f(x;\theta) = (\theta + 1) x^{\theta}$ with $0&lt;x&lt;1$ and $\theta &gt; -1$.  The likelihood function is: \begin{align} L(\theta) &amp;= f(x_1, \cdots, x_n; \theta) \mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{f(x_i ; \theta)}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\}\\ &amp;= \prod_{i=1}^{n}{(\theta+1)x_i^{\theta}}\mathbb{1}\{0&lt;x&lt;1\} \mathbb{1}\{\theta &gt;-1\} \\ &amp;=(\theta+1)^n \left( \prod_{i=1}^n{x_i}\right)^\theta \end{align}  We look at the log likelihood $l(\theta)$ and take the derivative of this since it is easier to deal with and it is allowed because the log is monotonic:  \begin{align} l(\theta) = n\log(\theta+1) + \theta \left( \sum_{i=1}^n{\log(x_i)} \right) \end{align}  $\implies \frac{d}{d\theta} = \frac{n}{\theta+1} + \sum_{i=1}^n{\log(x_i)} = 0$  $\implies \hat{\theta}_{MLE} = -\frac{n}{\sum_{i=1}^n{\log(x_i)}} - 1$  However, this doesn't look right. Why does Stephen Hawking say black holes don't exist?