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Graph of the electric field E(x) in some region is shown in the drawing to the right, and described bykeQ 3RZ E = keQ0 <r < RR <r(Don t worry about what ch...

Question

Graph of the electric field E(x) in some region is shown in the drawing to the right, and described bykeQ 3RZ E = keQ0 <r < RR <r(Don t worry about what charge distribution is causing this field )Write an equation for the potential V (r) using the reference V (0) = 0. Your function will be different forr < R and r you Il need t0 use curly brace to separate the tWO cases_ Hints: Be careful with your constants. Also, graph ofV(r) may help You to organize your thoughts and keep your sig

graph of the electric field E(x) in some region is shown in the drawing to the right, and described by keQ 3RZ E = keQ 0 <r < R R <r (Don t worry about what charge distribution is causing this field ) Write an equation for the potential V (r) using the reference V (0) = 0. Your function will be different forr < R and r you Il need t0 use curly brace to separate the tWO cases_ Hints: Be careful with your constants. Also, graph ofV(r) may help You to organize your thoughts and keep your signs straight.



Answers

The induced electric field (or eddy current field) is given by, $$ E(r)=\frac{1}{2 \pi r} \frac{d}{d t} \int_{0}^{r} 2 \pi r^{\prime}\left(r^{\prime}\right) B\left(r^{\prime}\right) d r^{\prime} $$ Hence, $$ \begin{gathered} \frac{d E}{d r}=-\frac{1}{2 \pi r^{2}} \frac{d}{d t} \int_{0}^{r} 2 \pi r^{\prime} B\left(r^{\prime}\right) d r^{\prime}+\frac{d B(r)}{d t} \\ \left.=-\frac{1}{2} \frac{d}{d t}<B\right\rangle+\frac{d B(r)}{d t} \end{gathered} $$ This vanishes for $r=r_{0}$ by the betatron condition, where $r_{0}$ is the radius of the equilibrium orbit.

All right. So it's considered are to be the radius with cylinder. The electric field e are due to the solid sonar distance are when the simple are less than our is less than or equal to the greater our can be defined by the following equation. So electric field off the small radius R ID equals lambda are over two pi times tentatively of peace face have the radius of the cylinder So this is just so this simple Ah, represents a small this is and the the large are here represents the radius. So, uh, so this is when this is are this less than or greater than the capital are And now for when it's smaller. So the other case this is true then e r is that equal to you, Amanda Times longer over two pi times primitive ity of peace base. And this time it's a simple are so on the surface of the cylinder potential B is zero. Now, if you move on to part a. So to calculate the electric potential inside the cell in urban, so inside the folder means, uh, the first. The first scenario, because it's inside similar. So the smaller radius is mess in the radiance of the sun. So let's consider the potential. So give our my next we are So that's the potential at the surface is equal to the minus, uh, into go off radius. So from the outer surface through the distance, all right, and then the electric potential e r in treated with respect to d r. So if you substitute uh for you are here, we're gonna get that equal to negative our And so we need to substitute the first situation that would spread the both such lambda are over to party e o times, uh, really cylinder square and integrated with Just respected er So if he integrated, we're going to get, uh Since this whole thing is ah, enforcing these are constants, You can actually bring it outside the the internal. And since our is the only one that's being integrated, all you need to do is to integrate that. So if you do that great, any kid, the final actor, as so these were the constant that can be moved outside the integral First, let me write that so and since it's just one are so just are over one. So if you integrate that, that becomes r squared over two. And if he also include the, uh, and points, then it will turn into so oh, are so it's r squared. Minus are? Yeah. So now, uh, let's go back into the main equation so you can actually simplify this further. Andi, uh, do it like this. Like looking to party primitive iti. And since this whole thing is, he waited by our recon Just if I divided since this this expression here is being divided by are here, so we can just simplify so it will become one minus r squared over a capital are square Sorry for that. Uh, they were called, but we know that BR the capital r is equal to zero. So from this equation here, this is a capital R. That's five that so therefore directly potential. So since media this is equal to fierce of them, this just equal the rz 40 Let me separate that. So this is just we are and it's equal to this. So this is electric potential inside the cylinder because we know at the surface electric tens of zero to this whole thing becomes zero. Just make sure that we know that this is equal to zero because of the potential there is that the surface of the cylinder there is no potential. So this is the electric potential when our isn't had seven. So now, to calculate the potential when on on the other case on the radius is outside the cell in there. So in this case, it would be Are So this case, second cheese Now you'd have to calculate the potential again. So what we are I must be at the lower is equal to negative they had to go from are to our And this time it's just Landau Two pi times primitively afi space time simple are which you're then interested of its respective er So now if they integrate this, we can move all of this. That means do that in a different color. We can move all of this. What side? The introductions There are constants and we just need to integrate one over Are exchanging alone are so now I fi right, The final answer. Get that lander over, Uh, two pi relativity Times a couple are And this this whole thing is times long capital are over our Yep. So and now we know that we are is equal to zero. So this is the potential. This is this is a mondo. Yeah, you can. So the electric potential outside the cell in there is equal to this. Sorry. There is a square here. Yeah, and that's it. First part A So the part B. So the graph of he as the function are given So here. Uh so this is the This is a function off he as a function of our. So let me draw that real quick. This would be at one are and to our and then it just keeps decreasing as the radius increases. And this the peak is at when it's equal to the radiant source. So this is E. This is the electric potential on this is the distance that you all know Onda. This is equal to as he described up here. So the grass off V as a function of our can be shown like this this this is electric electric fields. And now we're gonna do the potential. The potential can be shown by this following graph. So the potential here So let's do the do the x cornett again So that one raise two radius on three radius So one at zero it's gonna be two pi. So this is Venice departed from the surface and inside the 70. So the potential increases and goes to zero potential decreases goes to zero. And then now here we are at Lander so long to this negative black land along to over two pi primitive ity free space and let's do another one. So this negative lander this time it's gonna be a long three for three are so it's two pi times the primitive ity. So now we're gonna have it go down 23 are remember this potential against the radius and that's it for this question.

Allotments theatricals give it if they just started a job, just spit in with charges uniformly distributed in two words. What, then? In the first part, we have to fight you fought Party is greater than or equal to our. It behaves as a why Judge Concentrated had its center and B. But we have proof lined up what you should as a function of our for on its lesbian or people who are let us and start solving it. We have to use the content, but angel difference between the fights can be defined. Let's he, daughter Dia and second concept electric off the sword of the Spirit Thank you by our square foot are is equally important. And and he's you might. How do you square foot four? No start solving part. It's our little day. He put what each thank you by artists quit the potential ability minus. If you wait Artist square in trivia for the limit Infinity to part on solving it because you are so it just up. What? Can you shut up? Why judge que our distance rummage? So this is the interrupt A part? No, for audit capital of actually feel it if you wait artist quick four. Are you bite? Are U minus? Okay. Are the power pour upon our keep our foot? No, but you should be defined at minus some incident. You are the, uh you know aren't what he does. Dia as this is negative. No, you have it. Can you buy artist idea from a ticket? You late artist Quit. Or are you by capital Argue my mystery. Yeah, no on ordering it. We can like if you're a big thank you for your woman human get But my next school or directly we can write. I'm writing here directly after integrating, and that's keeping the limit you will get. Thank you for your are you over? But this is about off. People are Thanks for watching.

Welcome to the program number twenty two point five six. This is going to do go slow. So in discussion, we are given a sphere. And this fear has non uniformed job distribution and has ruled charge density, which is their calls to this. So for our less than our we have this much density and for our greater than there is no answer it. That means the charges contained within this sphere which has capital out ideas and were given that this quantities constant, which is Rome. Not so. First, we need to find out the charge contained it in the this volume. And then we will find out the electric field and then we will go into the DNC and Sadie and Park. So let's find out the charge. Condense. So we have jobs. Condemned his cue post Euro times, D V, which is the William Charge, then City times multiple volume which gives the total charge. So we have This is a sphere, so this would be drew on times. Davey Devi's equals two foot by our square DEA. So this gives us rule for by I'm square dia put in the middle of the road so this road goes from Jared. We're so we put a gelato so we have four by Rohnert. Gee, Toto are one minus for are over. Team Captain, our square Dia So doing the integration we have for by no note are too three minus for two years are for over food. So this goes away. So this is three and four goes away. So this comes out before by No no our Cuba with three minus our cube or three equal to Jiro. So total charge contained within this world is zero. But it is distributed over the hole sphere and we have minor sign. And this is something so that means some has negative charge and some has bought supercars within this whole but total jod is due. So that's your name not going to the be part. So we first we need to find our dialectic field at our president Capital. So we have go slow. So hee dude yes equals two Q and closed. What absolute! So we have kuen closed little charts contained within this warrior because we are talking about point outside the sphere. So this is our ocean surface now total charges we have already formed because of Jiro. So it is a prostitute and this is something to configuration That means we have e equals studio there. Feel outside. Is Julia not going to the spot? See, now we need to find an electric field. Are less than Capitola, so just e dor de s two enclosed. Well, absolutely not like toe. Let's take a look at the surface again. So this is a world capital? A Yes. Now we make a go ocean surface, which has ideas. Smoller. Now we have two enclosed, secure, enclosed would be true Davey the davies the area off this but one of this smaller sphere, which is it close to row for by our square DEA starting from Jill, Uh and we can put the value the role, which is this one. So this comes out to be food by and door Not our square DEA starting from zero to are minors Food cube over treated DEA integration, General. Oh, and that would be our protein and close to us. So whatever the quantity we have here, So we should replace with the quantity here. So Edo, the yes, Nadia's would be the area off the this fear, which is fourth by our square because two Q and clothes with this quantity. Well, we're absolute. So this gives us equals two four prior owner, this quantity over four by our square, you know, So we have this integration now are square. So let's integrate it out if we want to. So that's this disintegration will give us our Cuba over to me. And this integration will give us are for over to you. So this gives us the constants of four by four by goes away and this gives So the collective is articles, too? No, not over three. You know, times are one minus over. So that's your answer for your party. So their field within the sphere at distance are from the centre of the sphere now going to the but Dino Well, now we need to blow the let me feel worse is our distance. This is our This is our year. So if you'll take a look at the this graph and you can use the axle to blow the gaff or you can use, um, full line or maybe some program matematica to blow the grounds So the draft comes off you like this one. So the maximum is it calls to some value and we will find out the maximum in part. So that stood apart the family. So we need to find out the maximum this elected field. So let's take the daddy radio off. Let me feel and respect are so the maximum gifts is left that a video electrically with respect to our question, Jiro. So this one This is the condition for the maxim conditions. So this is the maximum useless The maxim This comes out to be Rohnert over T c Note to go or not, our mix t you know our equals zero the are Max is the point where the leftist realist Maxine his distance from center there in a book release Max. Therefore, this gives us our necks because too there were a lot of work to that means we have and the sender. That means that the halfway off this one, we have electric field Maxine, and this is left to feel we can find out. So put in the articles too are over too so equals to our two contributors Ruin our work city. You know what our two one minus one too. This gives us a one off over twelve, you know? So which is the cause of this maximally? So this is cost alone are kept locked. Porto. So that's your answer for this question. Thank you for watching the video.

All right, before we start this question that's going with the form for the electric fields for the difference. So if the radius is smaller than the Reedys, the cylinder and we have to use this following so that I could feel the heat. Of course, she is constant times the charge over the radius. Where and this is for the condition where the radius distance is less than the reason. So more than their we just listen here, all right to go over the next scenario. So for when there Reyes less than the radius, the formulae is easy to once again, Coombs constant times charge over the radius of the soon they're cute and this whole thing is multiplied by the expression ar minus three along a mix space over there. So it's minus three r squared before our So now we're going to part of the question. So to find out the potential, the radiance is greater than the radius cylinder. Do you use all integration from the Internets? Are two e the arbitrary point that we're completing from so are too big and we would use the equation that we just went over for the first condition. So it's cool. Sometimes. Charge where on that This is a respected TR, something akin to beat this. We're gonna count que over four i times primitively of the space. This is just expansion, of course. Carson times minus one over the radius from the intervals are to a so expand that further solving for the intervals It's cute times que over full high times Judy, I want to find cute. So a tensile. So therefore the attempts was I don't go to that produce right in charge in this region where the radius is less than the radius of the sun. So if you want to part B so this system for the potential in the other case, obviously for when the radius is greater than the ratings sunder. So in this case, the potential will be the integral From are today off the second form Have you went toe beginning? So let me just cook here. It's no wonder it again. It's a lot of expression. It's a lot of turns, so just will be down for this time consumes reasons. Now if you interviewed this, uh so I think we need to first expand this separated into the two into a well, since there is the radius of the cylinder between these into also need to make it 22 and at the two intervals get the complete control. So be radius from inside cylinder to the point to the surface of the similar. So I was cute times two hi across space times radius, and this is just expansion costs in again. We have the expression that played Jewish is AR minus three R squared over four for, and this is integrated with respect the D R. And if you add that into second intro are to a so the surface of the cylinder to arbitrate point outside, that's once against Syrians constant. Or you can wait as 1/4 pi attempts the productivity of space that's a radius with Let's just keep It cools constant for now. And this is times cute. You spared because this is where the second uh, since so this is for the second situation. If you saw that and into both these expressions, then we're going to get your feet, huh? Factory common turn to which is the charge of the group's constant. So the inspectors are first charged over pie that's a primitive, beautifully Spacehab radius. This whole thing is multiplied by R squared over two. So this is just the interval of this squared before and what's sorry? The name back in later. That's just r squared to Let's Get Simple are which is that enter into this far? Well, this is also headed chicken homes over at times. Q. And this is one of them because that's staying to go this. So now if you saw this, we're gonna get that the voltage potential is equal to cute. So we just expanding this sensitive. So it's just Q over two. I given space times are que four radius, and this is this expression is not to play by our cube to minors to a radius of seven returns. That's a finished


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