integrate-each-function-by-using-the-table-in-appendix-d-if-an-electric-charge-q-is-distributed-al-2

Question

Integrate each function by using the table in Appendix $D$.
If an electric charge $Q$ is distributed along a wire of length $2 a$ the force $F$ exerted on an electric charge $q$ placed at point $P$ is $F=k q Q \int \frac{b d x}{\left(b^{2}+x^{2}\right)^{3 / 2}} .$ Integrate to find $F$ as a function of $x$.

Vipender Yadav · Accepted Answer

So here on this problem we are given that the force exerted on a charge Q. Placed at a point P. Is given by if recalls to three times few times capital Q integration B D X divide by be square less X square. Hold the power 3/2. Now we have to find this F as a function of X. Now simply we have to integrate this and to integrate this, we&#x27;re going to use one formula which is given in the appendix T. And the formula is integration. Do you over you square plus a square. Hold the power 3/2 as equals two. You divide by a square. Underwrote you square plus a square and plus C. So this is a formula going to use. And if you compare the formula with the given in trouble we can say that we have U. Is equal to X. And do you will be equals two dx. And we have equals to be. So we plug the village here. Now you we have ex divide by a square means be square. Under route you square, S. X. Square and a square means be square plus seat. Now this constant will remain as it is, so we can put outside the bracket. So distinct. So this is the final solution for the given problem. That is the value of F. In terms of X. So I hope you have understood the problem. Thank you.

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.

Linh Vu · Answer

we&#x27;re going far, but where we have the inter go on the use sinew Do you ever get the gorgeous I knew? Why must you go sign new press A constancy in here We have a question off. Uh, thanks. Side Teoh X t x. So everybody knew you, could you That your thanks And then you culture patootie x So, I mean, it&#x27;s not just the exit, could you Did you are what? You and this accident a contender. You on with you. And now this is no way you could you could, you know, before we can remind us in the car in terms of, you know, funny, funny matures will be that you would you say you, Dean, you on do so that you cancel out? Yeah. Then we get Echo June into grow on the new sign You d you on my formal again? Incredulous. I knew when I asked you because you plus a constant C because you recruited your ex. So we have a side You. Thanks, Linus. Do thanks. Ghost site. You expensive Constance

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

Fuzail Shakir · Answer

hold the two out from the front. We lead to tend the integral of one over X square. Times of four plus three acts the whole thing square. So if we let you is equal to acts, so do you. Is he putting T X is that equal to four and B is equal to three? And if we then plug it in, we get two times the integral off one over you square times a plus for you, the whole thing square. Using the table into rules, we go to Appendix Speed, Formula 13. And if we plug everything in to Formula 13 we end up with negative too negative to over 16 times X plus. Sorry, that&#x27;s my ex Supposed before I was four plus six x Divided by X Times four plus three x plus three words plus six over four times Ellen of the absolute value of X, divided by four plus three x on dhe plus C. So if we simplify this a lot, we and with negative two plus three acts divided by four acts Times four plus three acts for plus three X minus three over 16 times ln of the absolute value off acts divided by four plus three acts plus C

Linh Vu · Answer

question Where you phone up? 51 Where were they? Into go under new ingenuity New teen you Then we get to go to the EU gone new numinous born pass a constant C and in discussion were given in to grow off the next labour Minister Angsty X So we applied in your culture minister to next. And then do you need a coach in the months to teach necks and therefore we can return This indigo here would be no. Now listen Means that X will continue over minus two. So accidentally could you did you over months to a about you Now the x ico June that the new over ones too. And therefore we can understand you know wanna go far Sign you, agent and you know you by formula getting Could you one of the phone times the able you humanist one plus a constancy Because you re coded a ministerial. Thanks. So we&#x27;re getting could be one of the four able minus two Thanks and then minus two minus one There&#x27;s a constant sick

Problem 52 Easy Difficulty

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Anna Marie V.

Samuel H.

Joseph L.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 28

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Anna Marie V.

Samuel H.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus