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A unifomly charged ring of radius 0.3 m hasa total charge of 150 pC Given Coulomb'$ constants k =9x 109N.m C ? The electric potential on the axis of the ring ...

Question

A unifomly charged ring of radius 0.3 m hasa total charge of 150 pC Given Coulomb'$ constants k =9x 109N.m C ? The electric potential on the axis of the ring at 0.4 m from its center equalsa) 1350 KVb) 75 KVc) 1.9x 10'Vd) 2700KV.Select one:dba

A unifomly charged ring of radius 0.3 m hasa total charge of 150 pC Given Coulomb'$ constants k =9x 109N.m C ? The electric potential on the axis of the ring at 0.4 m from its center equals a) 1350 KV b) 75 KV c) 1.9x 10'V d) 2700KV. Select one: d b a



Answers

A uniformly charged ring of radius 10.0 cm has a total charge of 75.0$\mu \mathrm{C}$ . Find the electric field on the axis of the ring at (a) $1.00 \mathrm{cm},$ (b) $5.00 \mathrm{cm},(\mathrm{c}) 30.0 \mathrm{cm},$ and $(\mathrm{d}) 100 \mathrm{cm}$ from the center of the ring.

In this question, we're told that a uniformly charged ring of Radius Sensei meters has a charge of 75 micro columns and were asked to find the electric field on the axis of the ring at various distances away from the center of the ring. So let's start off with a diagram here. So what we're gonna do is we're gonna take a look at the ring edge on meaning that the ring here you can kind of think of is coming into and out of the page. And we're just looking at, you know, a side view. So this is our ring. And let's take this as the center of the ring. And that means that the distance from the center to the outside of the ring here is 10 centimeters as given in the problem. And so basically what we're being asked to find here is the electric field at some points above the center of the ring. And what's being changed from eight a to B to C to D is the distance between the center of the ring and the point p. So basically what we can think of as the why value here what is the separation between the center of the ring and the point P. That's what's being changed. So what I'm gonna do is I'm going to find the electric field as a function of y. So I'm just gonna do, ah, general focus on creating a general equation for the electric field, and then we'll just be able to plug in the Y value that we're giving each separate part and will be, and we'll be done. So what we want to do initially is think about two little bundles of charge that are directly across from each other on the ring and think about how their electric fields will interact. So the electric field at Point P from D Q on the left will point up into the right and the electric field from Dick you on the right will point up into the left. And because of the geometry and the same symmetry of the situation, what's going to happen here is that the X component of D E, from the charge on the right will cancel out with the X component of D from the charge on the left. So the D E. E X is here are completely equal and opposite to one another. And so this is going to mean that the electric field from the total ring is not going to have any component in the X direction here. Because if you take any piece of charge on the ring, there's going to be a charge little piece of charge directly across from it that has an equal and opposite contribution to the electric field in the X direction. So if you're looking at the entire ring, the all of the little contributions of the electric field in the extraction are going to cancel each other out. So we don't need to worry about the electric field. In the extraction, there is none, and so we're going to concentrate on the electric field in the Y direction. So what is the electric field in the Y Direction from de que So the tiny contribution to the electric field in the Y direction from D. Q is gonna be equal to K D. Q. R. Squared Times Coast data and the fate of that I'm referring to is going to be the data in between the electric field vector and the vertical from that individual point. So this is the angle to which I'm referring Teoh. And so if we use co stato, we're taking the vertical component here of the electric field. So this gives us an expression for the electric field from one tiny bundle of charge. What we want is the total electric field from the ring. So what we're going to do is we're going to integrates all of the tiny little contributions to the electric field over the entire ring. Now, the good thing about this inter girl is that everything inside the integral is a constant, except the de que right K is a constant are is a constant. Every single piece of the ring is a distance r away from point P, so that doesn't change. And the angle is always the same as well, so everything can be pulled outside of the integral except the de que And so we're just integrating de que over the entire ring. So we just get the total Q Okay, so our expression for the electric field from the ring is gonna be K Q R squared Coast data Now, to make our calculations a little bit easier, what I'm gonna do is instead of using coast data, I'm going to rewrite this in terms of wise and ours. So let's take a look at our picture here. So if this angle is is data, then this one here is also fate up. And that means that this angle here at the top of this triangle is also data as well. And so what we can see is that CO state is actually equal toe. Why, over our and so we can simplify this formula even further to give us que que are over r squared times y over our. So we get cake. You Why over r cubed. So this gives us a really nice compact formula to work with when we apply the different Y values from part A too deep. So let's go ahead and do that. So this is the formula that we're using. This is the entire electric field from the ring. So for part A, we've got a why value of one centimeter. Okay, that's the distance from the center of the ring to the point that were interested in calculating the electric field up. And instead of one centimeter, I'm gonna use 0.1 meters. Everything should be in standard units. The radius we can calculate that's going to be given by Pythagoras theorem. So if you look at this triangle here, then we can calculate the are the distance. Ah, from the cue to the point p, we can calculate that using progresses the're, um so that's going to be at the square root of 0.1 squared. That's the radius of the circle plus 0.1 squared Thought is the Y value. And so that's going to give us a value of R of zero points. 100 four 05 actually will change that, too. Then it's important to keep those decimal places here because we are going to be cubing are so that does make a difference. So let's plug these values into our expression. So we've got that the electric field is equal. Teoh que que take you is gonna be the same for every single part here. So I'm going to write that down, especially separate. And then why over r cubed so cake you is going to be equal. Teoh. 2 674,050 That's just multiplying k by the 75 my curriculums. And then we've got a Y value of 0.1 meters and the R value Cute. And so that gives us a final answer for the electric field off 6.64 times 10 to the power seven Newtons per column. Okay, so we're just gonna repeat that exact same process for all of the different Y values that are given. So we've got next week. God's a value of five centimeters or 0.5 centimeters. All right, meters, rather and we're going to calculate the radius again. I'm not gonna write dot All out. It's a 0.1118 keeping a lot of decimal places again. And so the electric field is going to be 674,002. 50. Remember, that's que que and we're gonna multiply that by 0.5 divided by 0.1118 cubed. And so the electric field. The value for the electric field here is going to be 2.41 times tend to the seven Newtons per Coolum and for part C, we've got a Y value of 30 centimeters or 0.3 meters. The our value you can calculate. Then the distance between the the ring and the point is going to be 0.316 meters. And so the electric field is going to be 66 74 to 50 times 0.3, divided by 0.31 six. Cute. And that gives us e final value for the electric field of 6.39 times tend to this six Newtons per Coolum. So you notice that we're getting a smaller and smaller electric field as we go away from the center of the ring which makes complete sons. And then we've got for parts de. We've got a why value of 100 centimeters, a k a one meter and the radius is or sorry. The distance between the ring and the point P is about 1.5 and so we get a value for electric field and the final answer here will be 6.64 times 10 to the six Newtons per column. So this is our final answer for part D here

Now here the figure shoes. I'm just mention here that this particular figure shoes the uniformly distributed sighs that is small que across the ring. I would radius information as a no ask for the given information in step a step one. Roger. Ah, Okay, so first potential. But you stick in a sea of even and its values given us 45 syllables, next is the second potential. That would be re too. And value is 33 killer walls and the distance is given as eggs, which is 15 centimeters, which we would convert it into meters. That is 0.15 meter. No, just don't read. Step two. Here we are going to make use of the formula so opposes to find the electric. But I'm sure do to the charged particle. And it is given, as we is equal to K times Q. Upon our let's mentioned few things, your K stands for columns. Constant Q. Is the charge on the particle and our is a good distance from the particle? Okay, moving ahead here. The radius off the ring see, and so substitute a for are and the equation, which is that in step one so therefore even is equal to Kate. Times Q. Upon a. Therefore even upon K is equal to Q upon a. Now just think the reciprocal and square both the sides. So what we get here, let's understand. So even he's equipped to. So it is going to be a square upon que square is equal to gay square upon even square. So we took the reciprocal and we are squaring the same. So therefore, a square upon you square easy quit, too. Let's put the values here that is going to be nine into candidates to nine new gin Meter Burke, William Square, divided by 45 include tennis to three bowls, and we are taking the square offered. So therefore, finally, that we get E square upon que square is equal to four into canneries to 10. And let's mation this as immigration. Now for the Step three, I asked for the diagram. The distance off each small charge from the point on axis will be R is equal to on route off X squared plus C scream. We need to use mine to go steered him here for the more. Therefore, we two becomes que times Q divided by on route off X squared plus C square. Now square board decides so therefore what we get we do Square up on gay square is equal to que square Ah, born x squared plus c square. Therefore he squared plus x squares upon que square is equal to K square. Upon we do square we need to solve this further So a square upon que square Yes, X square upon que square is equal to he square upon reduce square therefore a square upon you square easy Quit you que square upon We could square minus x squared upon que square We are taking the term to the right inside so let's substitute the values here Therefore he square upon que square is equal to it's nine injured generous Tonight Newton Meters Square Park, Autumn Square the whole square upon 33 include generous to three world the host great minus 0.15 a the whole square and use graph on Let's mation this us inflation Be okay moving ahead Now compare the question a what be so what we get here therefore nine into tennis denying square I'm going 33 include tennis to three. The whole square minus 0.15 The whole square upon que square is equal to four into trendiness. Trittin therefore solving the 7.43 edge into 10. Minister 10 minus cto porn. Zero group to five upon you square is equal to four integrationist return. Therefore tree born 438 Interdistrict 10 is equal to 0.0 Kudo. Five. Upon que square it's vehicle Andy, but you need to solve this carefully. The Q Square is going to be 6.544 into generous to minus 13 and there's bad taking square root. We get eight form 089 into tennis to minus seven clumps. This is the answer that we have got, and basically, this is the rings charge brand. You okay? Next. Moving ahead for Steph, for we need to find rings Radius should. Therefore, it's substitute All right? Yes, subsitute the value off Q. In equation a off step two. So what we get here, that is a square is equal to four include tennis to 10 multiplied by Q Square, which is going to be four into a dentist retained, multiplied by value of Q is 8.89 include damages to minus seven columns the whole square and what we get here. That is zero for 90261 So finally, taking the square root is equal to 0.161 meter. So the us what we can right here that reigns radius is 0.16 meters and this is going to be the finer answer.

Well. And when X equals five centimeter, then electric field e is a cool too 300 into kill a Newton Burke alone. And similarly, when X equals 15 centimeter, an electric field E is equal to 160. Qilong Newton Curriculum Now Electric field E is he one by King Times Kinnel Times X, Divided by X Square plus a square, and we have three divided by to India Uh, exponent. Now let's substitute values now. This implies dead three 118. Multiply by 10 to the power three is equal to mine. Multiply by 10 to the power nine. Multiply by cure. Multiply by fight. Multiply by 10 to the power minus two, divided by 10 to the power minus six. Multiply by five square. Let's see screener uh, into three, divided by two. Let's say this is a question Number one and further we have 160. When exceeds, 15 centimeter is 160. Multiply by 10 to the power three is equal to nine. Multiply by 10 to the power nine. Multiply by cure multiplied by 15. Multiply by 10 to the power minus two, divided by 10 to the power minus six into 15 square less square to the power three by two. And let's say this is a question number to now dividing equity number one. And he couldn't remember to. So dividing Dividing Equation one. And do we have? Ah, 38. Multiply by tree into 25. Let's a square to the power three by two, it's equal to 16 into 225. Let's a square to the power three. By tour. So three by two and no taking two groups on both sides, we have tree divided by two into square. Rudolph, 25 lists. A square is equal to square root off. 25 225 225 less a square. No screening on both sides and spreading on both sides. We have, ah, mind. Divided before into 25 less. A square is equal to 225 plays a square and further weaken right. This is 200 25. Let's 90 square list mine. A square is equal to 900 less for a square right and five square equals 675 and A is equal to 11 point six to centimeter and severely cure charge. Killing is equal to one point 71 micro quilon

So he and I have already in the stead all the values, all the quantities in the problem or the norm. Quantities now for part eight when you defined the it to field at the Point X. So for that we lose the deduced expression in example, 21.9. So I have already waited in that expression. That question over here. So substituting each of these quantities over here, we find Electra Fear to be seven new temper cool apps and also hear capital Q is the charge on the rain, which is given to the 0.125 Lana columns on DDE. The value of K is nine times 10 to the bar. Nine Newton meters squared over columns quite know, make sure you convert each of these values into the standard units. So, for example, scenting a girl in tow meters on my no good lungs into columns. Now, for the second part, we have to find the force on the ring. Did you just charge you that displaced at X so forth? Let's find the force on cue. So for that, since we already know the electric field at Point X, which we deal. I've which recalculated in the first part so we can simply multiply that electric field with the charge that is placed over there on. Make sure you convert this micro columns into killed loans before substituting the values. So that gives us the electrification of the negative 1.75 times 10 to the minus five years on again, The direction the same asked the direction off the electric future. So that's I capped now, since the the force on the ring and force onto our action reaction forces. So the force, in the language of the negative off the force on the target killed. So that gives us the magnitude. Toby. 1.75 times 10 to the minus five. Neuter on the Todd. Sorry. On the direction is now opposite Tow the direction off the force on cue. So the election is positive. My cap


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