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$\bullet$ $\bullet$ Two particles having charges …

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Problem 42 Hard Difficulty

$\bullet$ In a rectangular coordinate system, a positive point charge
$q=6.00 \mathrm{nC}$ is placed at the point $x=+0.150 \mathrm{m}, y=0,$ and
an identical point charge is placed at $x=-0.150 \mathrm{m}, y=0 .$
Find the $x$ and $y$ components and the magnitude and direction
of the electric field at the following points: (a) the origin;
(b) $x=0.300 \mathrm{m}, y=0 ;$ (c) $x=0.150 \mathrm{m}, y=-0.400 \mathrm{m},$
(d) $x=0, y=0.200 \mathrm{m}$

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Physics 102 Electricity and Magnetism

College Physics

Chapter 17

Electric Charge and Electric Field

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Video Transcript

in this problem, we're going to be repeatedly applying this formula, which gives us the electric field, given the distances in the charge. And we're also going to use the fact that the electric field points towards the minus charges in away from the positive charges. And so let's start with party in part A. We have the situation that we're at the origin here. We want to find your lest you feel the origin to our right. We have kyu won and it's at a distance 0.15 meters and to our left we have another charge Q too also at a distance 0.15 years and in this case they're both native charges and so the electric fields will point towards them. They're being electric field here. You won due to cute one, an electric field here. You two do too cute, too because of the symmetry with the distances and the charges, we know that you want it's going to be equal to eat too. But because they point in different directions, I'm going to add a minus sign here moving the need to over I find that anyone plus he too is equal to the total, which is equal zero. And so, since they are in opposite directions, the results into electric field is zero. Let's draw the situation for Barbie, so we still have Q to here. You only have 21 here, and it's 0.15 meters here. It's another 0.15 meters here, and then it's another 0.15 meters to our point, which I'll call P, that we're interested in finding the electric field at. So now, the electric fields air going to point away. So Team one and that's you too. So let's figure out how they're magnitudes relate to one another. So using the formula on the first page there we have the Evil one is equal to 2,397 news for cool, and that's when we use 0.15 as the distance. Since that's assistance from P to keep one here, no e to we're going to use a larger distance. We're going to use three times 30.5, since that's the distance from P. Take you, too. And when we play with sin, we're going to get 266 news for cooling notice that the larger distance between P and Q two is causing the electric field to be much less than anyone to figure out the total electric field you total in the extraction. We're going to some, these two up, and we do that. We get 2,600 and 60 Nunes for cool, and it's in the positive extraction where this deposit extraction here. And so that's the answer to party. Fartsy is where the problem. It's a little tougher because we had to worry about the white component now. So we have que Tu and the one that say it's here directly below Kyu won is the point that we're concerned with I'ma call MP. And so there's going to be a lie component here of e to why, which is the white component? Watch for fuel do to this guy. There's also going to be an ex component electric field do to this guy e to X, and then that covers this guy with kyu won. There is no air, Moni, because P is directly vertical to it. But there is a lie component and so there is an e one. Why here and in total, the electric field is gonna point somewhere there. That's easy to hear, which is CTO. I need to ex combined and then we have to figure out what all three combined are. And so to do that, let's figure out what you want is even one again. We just are worried about the vertical component here. This distance, by the way, is point for d'oh meters in this distance here between cute too. In R point is 0.5 years and so even one can be found just like before because it's only American components. So we don't worry about the angles at all yet and we get that it's 337 Newtons per cool again. The one X is equal to zero. So Hee won. Why we'll run out of space here is equal to the full start here like that now e too, which is this diagonal vector here we can calculate like before and when we do that we have 216 Nunes for cool and now what we're going to do is calculate this angle here, the data and then once we have the data, then we can break this up into the horizontal and vertical components. Now you can find the fate of by using the a sine function, right? We know sign If they'd asked me point for over 0.0.5 and then we can take the inverse side of that. And when we do that, we get that data is equal to 53.1 degrees. This means that e to X is equal to e to times co sign a 53.1 degrees since the coastline gives us this horse on component and this is a being 130 noon speculum and then I'm just gonna write up here. You too. Why is going to be equal to minus E to sign theta minus because And to find my core consistently positive X and then positive. Why that way? So hee too wide points downward and so is negative. Also, that means I should have put negatives on these. So go and do that now, 53.1 degrees. And then now I'm going to plug in and I get negative 1 73 Nunes for cool. And so the total The toll in the extraction would be the sum of the different. He's in the extraction. And so this is equal to Yvonne X, those e to X, which is equal to 130 news for cool. And you, why is equal to anyone? Why plus E. T. Why? Which is equal to negative? 337 minus 173. She's going negative. Five ten 10 new ones for cool. Now that we have the components, we can figure out the overall direction that the electric field is pointing in. And so tangent of Phi is equal to Ugh. Why over yaks? I'm taking the house of value since I want a positive angle in the end, which is equal to 3.92 If you do that division there and then we take the understanding of this. To find out the overall angle, we get five people to 75.7 degrees. But really, this is the angle below. This is this angle here, so if we want to convert it into this angle, which would be how we traditionally right angles, then we need to take 360 degrees. It's attracted by 75.7, and when we do that, we get 284 looks like. So if you want to run the angle like this and be 284 degrees, if you're okay off writing you like this, it's just 25.7. Every way you do it, just make sure to report it. And then to find the overall magnitude of E, we need to take the square root of E X squared plus y squared, and we found these above here. Pulling those in gives us any value of 526 nudes for cool. And that completes Parsi Sting. So let's go ahead and draw the situation for party. We have Q wanting you to hear. I'm going, Tio not draw too much here because I think you see the idea here. The point. We're concerned it is directly above it in the middle like this. This is going to be an angle Zeta. And so that means there's an angle that here, too. I'm just not going to draw eveyone points upward like this. I should level the charges, at least so I ask, You wanna ask you to? This is the one and so there's going to be next component is going to be walking on it. There's also going to be you two here. So let's see Teo. It's going to be a vertical corner that in a horizontal one of that and these angles here r they did as well. So it's there in that state. And so let's actually calculate these electric fields so theta is equal to 53.1 degrees, and even one due to the symmetry of this problem is equal to me, too. And this is equal to you have to actually calculated using the formula for the electric field. It's equal to 100 63 noons per cool. Now E X is equal to Yvonne X. We'll see to X All right. It's just someone of this one here, and this one here, which is equal to zero. And the way to reason through this is that due to the symmetry, there's gonna be a component here pointing to the left. There's gonna be a component pointing right from this one's from Q two, and this one here's from Kyu won, but because of the symmetry, these have to be the same magnitude and they point the opposite directions so we can just reason that e X here has to be zero. No, you y is equal to eveyone. Why? Let's see to lie and even why and eating why will be equivalent again by century. In this case, they don't cancel. They combine. And so we get to and I'm just going to call anyone Time, Sign of data and a sign of fate is in their to isolate out that vertical component. And then the one is just that guy there. And so eveyone was found here to be 100 and 63. So we can plug in the state and we can float in the evil one. And we do that. We have 13 80 news for cool. Um, and so since e x zero, I can say that the totally the all that combined is 13 80. It's just the light component, and it's directly upward. And so you could say that final in the direction it's nine degrees or you could just say its upward, since the fine electric field is going to be pointing like that in that direction, and then it completes this long problem

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Watch More Solved Questions in Chapter 17

Video Transcript

Hugh D. Young

College Physics

Andy Chen

Zachary Mitchell

Jared Enns

Meghan Miholics

This textbook answer is only visible when subscribed! Please subscribe to view the answer

College Physics

Andy Chen

Zachary Mitchell

Jared Enns

Meghan Miholics

Hugh D. YoungCollege Physics

Andy Chen

Zachary Mitchell

Jared Enns

Meghan Miholics

Hugh D. Young

College Physics