Is induced EMF proportional to the square of the number of turns of a solenoid?

In order for this question to be clear I must first give some context:

Consider a 10cm long solenoid (solenoid A) of radius approximately 1cm with 400 turns.

Let the current in $A$, $I_A$, change with time slowly, $I_A = I_0(t)$ so that you may ignore the displacement current.

i) What is the instantaneous magnetic flux in the solenoid?


The instantaneous magnetic flux in the solenoid is given by $Phi =BS$.

The magnetic field $B$ can be found from Ampere’s law (ignoring displacement current):
$$oint vec B cdot dvec ell=mu_0I_{ell}=mu_0NI_0(t)tag{1}$$
where $I_{ell}=NI_0(t)$ is the total current due to $N$ turns each with current $I_0(t)$

The area is $S=pi times 10^{-4}$m${^2}$

Using $(1)$, $B,L=mu_0 N,I_0(t)$; the magnetic field $B=mu_0 frac{N}{L}I_0(t)$ where $L$ is the length of the solenoid. The flux $Phi$ is thus
$$Phi =BS=mu_0 frac{N}{L}I_0(t)S=4pi times 10^{-7}times 4times 10^3 times pi times 10^{-4}I_0(t)=16pi^2times 10^{-8}I_0(t)$$
$$approx 1.5 times 10^{-6}I_0(t)$$
which is the correct answer.


(ii) A voltmeter is connected across the solenoid. The meter will measure
$oint vec E cdot d vec ell$ integrated
along the wire. Derive an expression for the voltage measured (ignoring
the resistance).

By Faraday’s law the induced voltage per unit length is $$oint vec E cdot d vec ell= frac{dPhi}{dt}=-mu_0 frac{N}{L}Sfrac{d I_0(t)}{dt}approx -1.5 times 10^{-6}frac{d I_0(t)}{dt}tag{2}$$ So the total voltage is
$$V=0.1oint vec E cdot d vec ellapprox -1.5 times 10^{-7}frac{d I_0(t)}{dt}$$

This is the wrong answer and the right answer is basically
$$bbox[5px,border:2px solid red]{V=400oint vec E cdot d vec ellapprox 6 times 10^{-4}frac{d I_0(t)}{dt}}$$


On dimensional grounds, it was my understanding that the LHS of Faraday’s law, $(2)$ gives the unit circulation of the electric field, or, $unicode{x222F}(nabla times vec E)cdot d vec S$ (by Stokes theorem).

But to find the voltage $V$ across the solenoid we must integrate over its length, so that the units given by

$bbox[yellow, 5px]
{text{number of turns}times text{rate of change of flux} propto N times frac{dPhi}{dt}propto mu_0 frac{N}{L}Sfrac{d I_0(t)}{dt}times L}$

But in the box marked red the author is simply multiplying by the number of turns $N$, which means dimensionally the RHS of Faraday’s law is

$bbox[#F85, 5px]{text{number of turns}^2times text{rate of change of flux} propto N^2 times frac{dPhi}{dt}propto mu_0 frac{N^2}{L}Sfrac{d I_0(t)}{dt}}$

So no longer do we have unit circulation but more concerning is that the units go as the square of the number of turns. Is this really correct?

I checked Faraday’s law of induction so I know that $$mathcal{E} = -Nfrac{mathrm{d}Phi_B}{mathrm{d}t}ne -N^2frac{mathrm{d}Phi_B}{mathrm{d}t}$$
Clearly, I’m missing the point, so if someone could please explain it to me that would be great.

Program that turns on and off windows settings according to conditionals from cmd

I have been meaning to make a .bat that turn on or off some windows settings so I could bind it to a keyboard shortcut. I made a .bat file that attempts to add a registry key NoDesktop so desktop icons could be hidden, if it runs a check in reg and is unable to find the key value, or delete the registry key if it is able to find the key value.
@echo off

REG QUERY "HKCUSoftwareMicrosoftWindowsCurrentVersionPoliciesExplorer" 
/v NoDesktop 

if %errorlevel% equ 0 ((
REG DELETE 
"HKCUSoftwareMicrosoftWindowsCurrentVersionPoliciesExplorer" /v 
NoDesktop /f 
)& echo Registry key deleted, now the desktop icons should be displayed.
(
pause)
 )
else (REG add 
"HKCUSoftwareMicrosoftWindowsCurrentVersionPoliciesExplorer" /v 
NoDesktop /t REG_DWORD /d 1 /f (
& echo Registry key added, now the desktop icons should be hidden.
))

pause

but when i run it, the command prompt pops up and closes itself immediately. I tried then deleting all the parentheses but it shows a message saying the registry key is not found, which is good and should trigger the else command but it doesnt and it prompts me to exit.

what have i done wrong? i have also tried deleting all echo commands since I am worried that if a then b & c else d & e doesnt work in cmd. but it still shows that the registry key is not found and prompts me to exit. I have also tried tampering with parentheses and make sure they are all closed but its still not working.

What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear…

I’ve read a lot about PCA, including various tutorials and questions (such as this one, this one, this one, and this one).

The geometric problem that PCA is trying to optimize is clear to me: PCA tries to find the first principal component by minimizing the reconstruction (projection) error, which simultaneously maximizes the variance of the projected data.

enter image description here

When I first read that, I immediately thought of something like linear regression; maybe you can solve it using gradient descent if needed.

However, then my mind was blown when I read that the optimization problem is solved by using linear algebra and finding eigenvectors and eigenvalues. I simply do not understand how this use of linear algebra comes into play.

So my question is: How can PCA turn from a geometric optimization problem to a linear algebra problem? Can someone provide an intuitive explanation?

I am not looking for an answer like this one that says “When you solve the mathematical problem of PCA, it ends up being equivalent to finding the eigenvalues and eigenvectors of the covariance matrix.” Please explain why eigenvectors come out to be the principal components and why the eigenvalues come out to be variance of the data projected onto them

I am a software engineer and not a mathematician, by the way.

Note: the figure above was taken and modified from this PCA tutorial.

How does ‘The Barkeep(Agent Doe)’ turns Fizzle Bomber in Predestination?

In the movie Predestination, one of the pre climax scene shows that The barkeep(Agent Doe) turns as Fizzle Bomber by killing the future him. Please find the Wiki link below:

1975-Mar-06 Agent Doe kills the Fizzle Bomber, his future self, thus becoming the Fizzle Bomber.

But Agent Doe is the one who actually tries to disarm the bombs and saves lifes. How does a guy such as him(who saves lifes) turns as Fizzle bomber(who takes lifes)?

Difference between the verbs 轮 and 轮流 (meaning to take turns)

In HSK 5, lesson 1, the word ‘è½®’ is part of the word list, with the translation: ‘v. to take turns’. In lesson 3, the word ‘轮流’ is part of the word list, again with the translation: ‘v. to take turns’.

My teacher has told me that there are three differences between these two words:

  1. 轮 is informal (口语) and 轮流 is formal (书面语)
  2. You can’t say 轮流到你了, but you can say 轮到你了(it’s your turn)
  3. You can’t say 轮去做.., but you can say 轮流去做.. and 轮着去做.. (taking turns doing something)

This still doesn’t really explain why you can’t say such a thing and what the difference actually is, is there somebody who knows?

Column data turns to NULL after Encryption

I have A table

Customer_data (Customer_id,Customer_Name,Credit_card_number,Credit_card_number_encrypt)

What I aim to do is to Encrypt “Credit_card_number” and insert it into “Credit_card_number_encrypt” I’ve followed the instruction of creating MasterKey,Certificat and Symetric key. The problem is that when after Encrypting the data the content of the encrypted column turn to NULL !

Select EncryptByKey (Key_GUID('SymmetricKey1'),Credit_card_number)

from dbo.Customer_data

enter image description here

this is how I’ve created the symmetric key

CREATE SYMMETRIC KEY SymmetricKey1 

WITH ALGORITHM = AES_128
ENCRYPTION BY CERTIFICATE Certificate1;

Question: 2005Volvo isx Cummins 475 my oil warning light turns on and the motor shuts off but my oil pressure…

My oil gauges say that the temp and pressure is all normal but the engine shuts and the oil warning lights turns on any one know why that is I’m stuck on the side of the freeway