solve-the-given-problems-by-integration-if-an-electric-charge-q-is-distributed-along-a-straight-wi-3

Question

Solve the given problems by integration.
If an electric charge $Q$ is distributed along a straight wire of length
$2 a,$ the electric potential $V$ at a point $P,$ which is at a distance $b$ from the center of the wire, is $V=k Q \int_{-a}^{a} \frac{d x}{\sqrt{b^{2}+x^{2}}} .$ Here, $k$ is a constant and $x$ is the distance along the wire. Evaluate the integral.

Vikash Ranjan · Accepted Answer

in the problem we have been given the equal okay, q Integration of -8 to a dx upon whatever of B squared plus x square. To solve the problem we will assume x equal be done theater The X equals B six sq Tita de Tita. Well, first survey in different integral that is taxable. Sorry that is the ex upon road. Uvarov the square plus X square This is regional integration B six sq Tita de Tita upon. But under The Squire one plus dance party died. This is equal to be integration six Party to deter to upon B and here it is sector to this becomes integration shake theater Digital which is equal to Ellen mod Sector to plus 10 tita plus C. Now further this is written as Ellen mud X by B plus whatever off one plus X squared plus B squared. This is equal to Ellen model X five V plus right Uvarov, B squared plus X squared upon B and put the limit that is minus A to a. To put in the limit we have this equals two. Ellen mod A plus right Uvarov A square plus be a square upon minus eight plus. Cover off a squared plus B squared. So further we have finally, V equal que que el in mud A plus Todorov, A squared plus B squared upon minus a plus. Suvorov esquire plus B squared. So this is the final answer to the problem.

Mayukh Banik · Answer

they&#x27;re set up the problem carefully. Here we have Y axis. And here is our X axis. The rod that we&#x27;re concerned is extending from negative as a war to toe positive. As aware too, on our point is here P which is an X y. So the electric field. So there has first choose our little cedar at a distance off L electric fields. Goto this say that would be in this direction. They restrict the horizontal and the vertical component, which is if this act is theater. This is E Hussein data, and this is E scientific. Now shot on this leader is De que equal total charge over total length times the l. This angle is set up, so e x is Yeah, yeah, yeah, Actually, that&#x27;s just fine. E forest total charge Over here e equals que times dick you over distance square, which is X minus l square plus y squared. Now we have Hussein theta equals This is point X comma. Why so concentrator is the base which is X minus l divided by swelled off. Explain SL square plus y squared on scientific A will be why over square root explains l squared plus y squared. Mm. So finally, we have total electric field e X equals G. So this is E Hussein theta integration equals Mm hmm. Que que over l times t ell over x minus l times so de ico scented x minus L times d l over explain is l squared. Thus y squared three over to on similarly e y with the integration be scientific data. Oh, uh oh. But right after doing the integration, you I will come out to be just negative. Que que over wife.

Luis Rios · Answer

give this problem This saying that between points p one p two the radius this constant. So that potential do you want to be too is the same. And this is what we&#x27;re gonna integrate over this. This is the path. So then our potential between points A and B Let&#x27;s call it is into girl A To be in an electric field is what&#x27;s given Al fall over s and DS. The inner Integral becomes Alfa No and yes, in the interval A to be so Then just expand that and you get helpful. Ellen of B minus currencies l A of a. So then are into more than our enter for the potential becomes how far? Well, then be my ovary.

J Hardin · Answer

a charge through out of length. L produces an electric field at the point p, given by the following. Integral. And here&#x27;s a picture of what&#x27;s going on we have Here&#x27;s our Rod and Blue of length l There&#x27;s a point Pete here. We just need to evaluate the integral and just find another expression for the electric field. So really here. Let&#x27;s just try to go ahead and use the tricks up and just evaluate this integral. See what we get inside the radical and the denominator. We see X squared plus B square so we can go ahead and choose X equals B tan data. That should be our tricks up. Taking a differential on the side d X equals B C can&#x27;t square. Then let&#x27;s go ahead and rewrite this. First. Let&#x27;s actually notice that we can simplify the denominator X squared plus B square Scott and simplify this now. So we had B square tan square plus B square factor about that B square tan squared plus one in the parentheses. And we know that sequins were and then we&#x27;LL go ahead and raise this the three house power. So let me write this over here somewhere. So here will just go ahead and multiply too. Two, three halfs. And we just couldn&#x27;t be cubed seeking cute. So let me go to the next page and simplify this integral. So originally, let me write the original Integral. That&#x27;s the land of B for pie. Absalon Subzero X squared plus B square to the three halves the ex. Now, after we do a tricks of these limits of integration may change. We don&#x27;t know what they are yet. Let&#x27;s just call them fate a one and a two. And then we could actually go ahead and start factoring out. So let&#x27;s simplify. We have let me actually make some more room here, pull up the constants. We have pi times bees going Pull the bot for pie. Absalon, Subzero. So, basically, just pulling this out just a constant. No exes there. Then we have the integral. They don&#x27;t want to stay, too. We pulled out the Lambda, so we&#x27;re left over with the eggs, which is B sequence, where data data and on the bottom, we&#x27;re just left over with. We simplified this in the previous page. It was just be cute seeking cubes there. Now, here we have b squared on top. Be cute on the bottom. So we&#x27;re left over with one over b. So that will give us a B here in the bottom. And then we&#x27;re just left with the integral fatal one data too. And then we just have one over C camp. We know one over. Seek it. This is just coast on data in the general. That is your sign. Spread those constants absolutes of zero. And then we have the integral co sign, which is signed. Then we should technically put our Fada one and data to. But since we don&#x27;t know what those data values are, we can draw the triangle, use the triangle to go from sign back into the variable X. And then we could use our original limits, which for? Minus a and L minus a. So here, Lester on the triangle. We had originally X equals B tan data. That was our tricks up. Sulfur. So here&#x27;s our data. So tangent opposite over a Jason X over b Call the hype on his age. Then by Pythagorean serum we have a square is X square plus B square and then solving for age. It&#x27;s square room escort. These four. So now sign is just X divided by H so we can go ahead and replace scientific with this expression here. And then we can go ahead and go back into our original limits. I&#x27;m going to say and minus a So let&#x27;s go ahead and write that on the next page. You still have our constant out here. And then we evaluated, signed back in terms of excusing the triangle and then were a lot to go back into our original limits since we&#x27;re back in the Variable X and then out of the last thing to do is just to plug in thes leper. Lower and upper limits is X. There&#x27;s our land. Ah, for pie trio Excellent, Obie. And then plugging in l minus a. We have l minus a up top square room, a minus a square plus B square minus. But then I&#x27;m subtracting on minus eight or so. Let me go out and actually turn that into a plus. Because of that double negative, Then I have a A squared plus B squared in the bottom on the radical. And so this is a different expression for the electric field. So really, we just use the trucks have been evaluated the integral, so I would stop right here

Athiru Pathiraja · Answer

all right. So it&#x27;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&#x27;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&#x27;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&#x27;s inside similar. So the smaller radius is mess in the radiance of the sun. So let&#x27;s consider the potential. So give our my next we are So that&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s r squared. Minus are? Yeah. So now, uh, let&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s just Landau Two pi times primitively afi space time simple are which you&#x27;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&#x27;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&#x27;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&#x27;re gonna do the potential. The potential can be shown by this following graph. So the potential here So let&#x27;s do the do the x cornett again So that one raise two radius on three radius So one at zero it&#x27;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&#x27;s do another one. So this negative lander this time it&#x27;s gonna be a long three for three are so it&#x27;s two pi times the primitive ity. So now we&#x27;re gonna have it go down 23 are remember this potential against the radius and that&#x27;s it for this question.

Khoobchandra Agrawal · Answer

hello Breaks in this problem. It is given that while you leader in a victory, do you know, sitting down having the judge and deflection off are both inside and outside. They have to Gallipoli, the rotation bientot fire the pain till inside and outside you like and a victory off birth surrender. And be part we have to learn the Rob become potential and electrically as a you know, uh, what already he&#x27;s from you Don&#x27;t know three yet. Let it start solving has been electrically inside the surrender in Lambda. I just, uh Then there are upon who by absolutely not artist square for our is listen or equal to our no and upset that slender people in lend up on food fight. Absolutely not Our foot are these better than or equal to capitalize the relation between electrically and potential. Its potential difference is defined. It&#x27;s eat or dear. So this concept will be able you to solve this problem. We have to use this one sort of it. No, it&#x27;s backside. Me first put upside and ticket but inside west get what I would say. Remember, it is given what? The intelligence that place is zero and didn&#x27;t put that site. Oh, are what? And your potential is zero on It is so integration are left in sight. Djotodia, we will absolutely very well be outside this damage. Constant indication off Dhanapala will be No, no Are so but Intel difference. You will get land upon by Absolutely not. No, no, it&#x27;s more art. Capitola, this is for outside What R is greater than capital our own. All but inside What? Inside the surrender. That is our It&#x27;s less then. Oh, that you were given doodle on simplifying it. You will get left up on board bite. Absolutely not one minus. This is on their part. But a child inside human right inside the sender potential is for our it listen or equal to are so this is unsettled are a no we have toe brought a grab support grab electrically after center off The that is at the access video access to that it included on becomes maxim or person lend up on who by excellent not are that it started depleting inversely proportional to the States and becomes deal work and pretty But they make it back some after certificates study on the exit as people from extras to surface. It&#x27;s political, is it? And becomes a minimal that is You don&#x27;t on that street. Then it starts, including So these are going to wonder for party. Thanks for watching.

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Solve the given problems by integration. An elect…

Problem 37 Easy Difficulty

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Anna Marie V.

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Kristen K.

Precalculus Review - Intro

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus