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Four charges are placed at the corners of a square of side a 1m, with G1 = Gz =-1C, Q=q4 =+1C. Initially there Is no charge at the center of the square: The Coulomb...

Question

Four charges are placed at the corners of a square of side a 1m, with G1 = Gz =-1C, Q=q4 =+1C. Initially there Is no charge at the center of the square: The Coulomb constant is 8.98755 x 10"N.m'/c' .BEFOREAFTERSa) Find the wark required to bring the charge (rom infinity and place it at the center of the squareSb) What is the magnitude of the total electrostatic energy of the final 5 charge system? Use Scientific Notation for answverl

Four charges are placed at the corners of a square of side a 1m, with G1 = Gz =-1C, Q=q4 =+1C. Initially there Is no charge at the center of the square: The Coulomb constant is 8.98755 x 10"N.m'/c' . BEFORE AFTER Sa) Find the wark required to bring the charge (rom infinity and place it at the center of the square Sb) What is the magnitude of the total electrostatic energy of the final 5 charge system? Use Scientific Notation for answverl



Answers

Four point charges, $q,$ are fixed to the four corners of a square that is $10.0 \mathrm{~cm}$ on a side. An electron is suspended above a point at which its weight is balanced by the electrostatic force due to the four electrons, at a distance of $15.0 \mathrm{nm}$ above the center of the square. What is the magnitude of the fixed charges? Express the charge both in coulombs and as a multiple of the electron's charge.

In this problem we are given for charges all the same attitude to positive to negative. They're to be placed at the corners of a square such that the force in any charge it toward the center of the square. The configuration I've shown here actually does satisfy that you on your own switching just switch minus and a plus. And you'll find that you do not meet that criteria. Now, before we get into looking at the forces, we do need the diagonal distance. This is a square. So it's very easy using pythagorean theorem to get that the diagonal distances L A times square with it too. We need that in a minute. No. Let us start looking at the forces and I'm going to use charge for to look at the forces on. And I've noticed as I've enabled labeled each of the charges 123 and four. Let's look at charge one on four minus plus. That's going to be attractive. So that is 41 two, four plus plus as repulsive because I'm using the grid. I'll explain more about that in a second. This is for two three on four minus plus. That's attractive. Okay so we have our three forces on that. Now let's talk a little bit about the grit. One and three are all distance away and all the charges have the same magnitude. So the magnitude of the force is exactly the same. 4143 are exactly the same. Now charged to though is L. Transcript or two away farther away. Remember there's one of our square and columns law. So this forest magnitude will be less. So notice the arrow. The arrow on 42 is less in length than the arrows on 4143 That's very important for what I'm going to do next. This problem really is set up to be done graphically. Doesn't have you could obviously could still do it F. F net except net wide. Break everything at the components and do all that work. Nothing nothing wrong with that. But this one is also very destructive to reinforce the graphical factor concept. So let us add these vectors up graphically and that's going to use the triangle rule. Well we so we take this we want to add 43 and 41 So we're going to take the tale of 41 put the tip of 43 So here is for one redrawn so its tail is at the tip of 43 That I draw from the tail of 43 The first factor to the tip of the second factor which is 41 That factor is 43 plus four one at vector some that's not the net yet. But that's the sum of those two. Now we've got 42 still in here. So now we do the same thing taylor 42 to the tip of the 4341 Some. So I'll put that right here so that is for two be drawn. And then they get the sum of all to get to some. I draw from the tail of the first which is a 4341 vector the tip of the second which is the 42 factor. This factor is the sum of all three. That is F net at his F net which is F 43 because F 41 plus F four to So that's done purely graphically. That's why I need the grid. Now notice it is a long look at the grid. All these vectors are along the diagonals. So we meet in the F net is pointing toward is along that diagonal towards the center. So we meet the criteria again, switch these two like this positive, make it negative and you'll see that it does not work. Now. The purpose of this question the end was to get the magnitude of F nut. How do I do that? Do I still gotta go break it up into components and all that stuff? No, I still use the graphical aspect of it. So I'm looking for I'm looking for F net the magnitude. Now now when we're along the same line, think about having two vectors along the X axis. One in the positive acts one of the negative backs and that's it. They have no other components. So what do I do to get the magnitude of that some I'm going to take the positive that say the positive ones larger. Positive one minus the one point in the negative. And I'm done. So don't let this being on a diagonal bother you. I have I have this 4341 vector along this line. And the four to vector it's along this line. So think of it as a plus diagonal and mona's diagonal directions if you like. Just like you have plus action negative acts exactly the same idea. So F net is the following man two of the 43 because for one some minus the magnitude of F four to that gives me the the net the magnitude of the net force. Now, where do we go from here? Well let's look at what the magnitude of this. F 43 F 41 is. Well aren't 41 and 432 sides of a right triangle and they're the same length. So this can be rewritten as F 43 square plus F for one square minus F 42 And I told you that 43 and 41 And hear from the diagram are the same. So I can factor that out. I have 43 squared plus four three square. So that's an F 43 scramble to to minus F 42 So now and I'm going to give myself some rooms, let me come down here. Now all we gotta do is put in columns law. So 43 it's going to be K ties. The magnitude of the first charge has made to the second charge well, but their managers are all the same. So this is Q squared. The distance between three and four is L. So this will be L square on the bottom ties. The scripture too. Don't forget that minus. Okay, now four to get both charges, You have the same man. Too cute, So cute. Square. And we have here, L scripture too squared. And you can see KQ squares in both terms. L square is going to be in both terms such factor those out. Okay, Q squared elsewhere. We got the scripture to still left over in the first term. And now let's remember we had square to squared in the bottom. So it's going to be minus one half. So now we can put in our numbers and get if not the magnitude. So 8.99 times 10 to the nine newton meters squared. Cool. Um, squared. They've told us that it's to Michael Collins for the charge. Two times 10 to the minus six columns. Don't forget to square that. Mm mm. And then we have the script to minus the one house. And then we got square on the bottom, which is 0.3 m square That and this works out to be 0.365 new tits, so that's the magnitude of the net force and that is the whole problem. Mhm.

Hi in the first part of this given problem there is square And all the four corners of this square are carrying identical charge, identical charges, each having a value of plus or 0.0 micro column. If you consider these vortices will be A B, C and D. Then net electric field at the center. The center which is the point of intersection of the diagnose this. Let it be all using the concept that electrical goes away from the positive charge. So There are four electric at the center. one because of the charge put at a going away, another because of charge put at B and again going away then carbon due to the charge products see again positive. So are there in the last one you do that put a deep again positive. So going away and using these electricals, R E A, E C, E B and E B. In this first part of the problem, each charge is 4.0 Micro Coolum and the stance of observation point, which will be half of the length of diagonal and diagonal. Here is Side route to side which is given as 20 centimetre into route to Divided by two. So the distance of this observation point from each charge that is Then route two cm. Then using the expression or electric field Okay into Q by artists where we conclude as the charge Q and distances are are all the same and the directions are opposed. So we conclude E A is equal to E B. It's all too easy. And that is Eddie. So we conclude E E will be cancelled by E. C. And E. B will be cancelled by Eddie. So net electric field data center Will be zero, similarly In the 2nd part of this problem, in part B. Then the opel is square carry identical charges. Here this is plus 4.0 micro column. This is also a plus 4.0 micro column. This is -4.0 my curriculum, this is also -4.0 Micro Hula. Well now electric fields at the center again they are equal and opposite. This is A. B. C. And D. So electric fields away from the positive charge. This is E. A. That will be cancelled by easy and electric fields towards negative charge. This time it is here he sees up but again a positive direction. So again net electric field at the Centre zero. But in the her part of the problem the situation is a little bit different. The same square same charge particles. Same diagonals, magnitudes of the charged particles same. But this time their locations are different here. This is plus 4.0 Micropal. Um Here this is -4.0 Micro column. And here Here this is also plus four, This is -4.0 my curriculum. And here this is -4.0 Micropal um at A B&C. & D. So if we look at the center this time the directions of electric field away from positive charge and towards negative charge. So in the same direction they're also away from positive charge, but towards the negative charge. So again in the same directions The two electric fields. So here this is E. A. And E. C. And here this is E. B. And E. D. and having a 90° angle between them. So if you consider the addition of the school officials to be E one and these two P B E two, then even will be equal to E. A. Plus E. B. Or we can set twice off E. A. And that will be given by two into K into cuba are swell. Yes, this will be do in 29 into 10 days per nine into charge for my curriculum or 14-10 days for -6 column divided by It's quite a distance which was 10 cm or 0.1 m into route to this electrical finally comes out to be 3.6 into And Part six Newton Park Coolum. And similarly The same will be the electric field E two. Now angle between them That is 90°. So net electric fuel here at the center vector edition of even And E two using Pythagoras. Tehran as even and the two are also same. So we can say This is E one E square plus this is also even the square Is this is two times off. Even in square or we can say this is even do. So This is 3.6 in two into standish bar six Newton purple, um, or finally net electric field at the center of the square newton Park column. Or we can say this is The net is equal to 5.1 mega newton per column, which is the answer for this given problem here. Thank you.

Okay, so if we have the four charges Here we have -1 Micro column church At this corner, we have two micro collapse. Here we have minus three micro column. And here we have for Michael klump charge, this is a square and these four charges are placed at the corners. So the square has side off length five centimeters. So let a denote the side of the square and it is five centimeters. So let's convert all of them in a C. Unit. We get the side as five times generous to minus two m. And here we need to find out the force on this to my curriculum. So we need to find the magnitude and the direction. So first the force and to my curriculum will be due to all these point charges. So do 2 -1 Micro column. It will be attractive due to four micro column. The force will be in this direction and do two minus three micro club. The force will be in this direction along the site. So let us see these forces F. one. These forces have to and these forces have three. So we will consider the coordinate system. Mhm. Yeah. The coordinate plain ex wife. Mhm vertical axis representing by access and horizontal representing the X axis. And here F one can be computed using columns law. So we used the column slow here. Okay. You want you to buy our square and the value of case 19-20 part nine. So F one will be K times the product of two charges will be two times one. That is too. But they are in my curriculum. So we will get 10 ways to -12 over the square of the distance separating them. That's the square of the site. So it will be five times 10. Raise 2 -2 whole spread supporting the value of key. We get everyone here has nine times 10. Raised to nine times two times 10 ways to -12 over 25 times 10. Rescue -4. So this gives us the value of F one as 18 x 25 and this 10 to par minus four, intend to par minus store, gets cut, we get 10 to par minus eight and 10 2. Part nine -8. That will be tend to part one. So it will be 185 25. So let's simplify this weekend. 36 x five. So we write the value of the force in the simplified form that is 36 by five newtons. And since this forces a vector quantity, we will add the direction as well. It's to the left. So this forces minus statistics by five. I can So we get F one. Similarly we get I have to hear So after can be computed similar to the way we computed f. one. So after will be okay so will directly put the values now we already know the values Q. And Q. Two. That means two times three. That will be six times 10 raised to minus 12 Over our square. That will be 25 times 10. Raise to -4. So we simplify this and we get This has 54 by 25 Times 10. So this week um 108 5 5 Newton's after his 108 5 5 newton's and this is along the negative Y direction because it's going down so it will be minus this Jacob. So now we will find the magnitude of this f. three. So after we now also can be calculated in the same manner. So this is the first between two micro column and for my curriculum. So it will be eight times Cuban Q two that is eight. Thanks to generous to -12 over 25 times 10, rest -4. So this comes out to be 7 25 25. So this will be 72 into 10 x 25. And this becomes This gives us the value of F three, The magnitude of F three us 1 44 x five Newton. But if he observed this extreme We are seeing that this F three This 4th F three is acting along the diagonal, right exactly along the diagonal away from it. So we find we take its component along The x axis and y axis. So the angle here is 45°. So we project this force F three force will be 1:44 x three Across 45. Its component along x axis icap Plus 1 44 x three signed 45. Hello Jacob. So simplifying this gives us the value of F three as 144 x three route to I kept plus 144 x three route to check up. So here we got after three. Now we needed to compute the net magnitude a direction of the force on this. To micro column. Pleased at this. Why so to do that we find the Net force and the Net force would be The resultant of these individual forces have one vector plus have 2-plus 3 victory. So we add them. It comes out to be -36 x five. I kept minus 108 by five Jacob and plus 1 44 by three route to I kept plus 1 445. three route to Jacob. So let us combined Icap and Jacob together. So this comes out to be 1 44 by three route to -36 x five. I kept plus 1 44 by +32 -008 x five Jacob. So let us compute in terms of decimals here. So the net force here when computed in the simplest form that will come out to be mhm 101 eight minus 36 x five. We do. That is 7.2 I kept bless 101 it minus 21.6 Jacob. So this comes out to be that the net forces 94 point six I tap and then 80 point Jacob. So from here, once we get the net force we can find the magnitude the magnitude comes out to be some magnitude of net force. That is definite. This will be square root 94 point six whole square plus 18.2 whole spread. So that comes out to be square root. Mhm 15381 point two. So if you take it square root, this comes out to be uh huh. 1 24 Newton's approximately. So we have got the magnitude of this net force and the direction of this force that direction with respect to horizontal. It can be computed by taking the angle As 10 universe or by component that is 18.2 over 94 0.6. So if we do 80.2 over 94.6 and take the 10 universe of this mm this comes out to be 40 point three degrees. So we have figured out the magnitude As 1 24 Newton's and at an angle of 40.3° with respect to the horizontal. Now, in this same problem, let's say we want to find out the value of the electric potential at the center of the square. So this was the square and the charges that were Places to Micro Colombia -1 μ Colombia for micro Colombia. And sure we had -3 My curriculum so we need to figure out the potential at the center. So we will use the expression of potential that is cuba. And assuming that the value of K. Here will be put as it is present, not the magnitude. So this is a scalar quantity. So we will just add the potentially to all these four charges. So let us say the potential you do To my curriculum will be we won due to -1 v. two. This is the three and 34 and its distance from each of these charges to the center. That will be seen because we have the side us point, Let's stick in Powers. So this was 5-10 ways to -2 m. So this also was 5-10 race to -2 m. We took the side of the square and it's stagnant will come out to be 5, 2 into 10 race to -2 m. Using the fighter. First film, we get the stagnant and The distance will be this diagonal, what we found and divided by two. So each of these charges will be placed at a distance of white By two into 10 days to -2 m. So here we compute this. We won. We won will be K times cube. That is two in 2. 10 ways to minus six by our that is five by route to In 2. 10 ways to -2. Similarly, we too will be minus K into One into 10, raise to -6, divided by five x 2 into 10 raise 2 -2. Similarly we figured out to be three and movie for so we three will also be K times four times 10 raised to minus six by five Bible to into 10 ways to minus do. And this will be minus K times three times generates two minus six over five x 2 times 10. Race to -2. So the total potential at the center that will be given by B one plus B, two plus three plus. Before and this we put the expression here and try to simplify a bit So that this -2 and -6 in all of them, it will be cut and it will become Tend to part -4. So key value will put us 9-10-5, 9. So if we simplify we're gonna get 18 Route two x 5 times. Yeah, 10 ways to price. And we too will be minus 9, 2 x five. Enter 10 Ways to Fight. Similarly with three will be 36 55 route to into 10 ways to fight. And we four will be also a negative form. So 27 2 x 5- 10 raise to five. So we take this tendency to fight as the common value and We do 18 to buy five -9, 2 x five Plus 36. route to buy five -27 to buy right? So simplifying this value, we get no 18 to my fight. So this can be approximated as 5.1 into 10 raise to five. But so this will be the total potential at the center. And finally we figure out let us consider the configuration again. So we had to micro plum here minus one microphone for micro column and -3. My Brooklyn and now we will figure out what is the work that is to be done to move a church of magnitude let's say one Nana Colombia Which is 10 to -9 column from infinity two center. So we are bringing a charge from infinity to center. What will be the work then? So we have figured out the potential at the center that is 5.1 into 10 raise too. Fight report and we know that the potential at infinity will be zero vote. It's relatively taken a zero because the separation increases to infinity. So the potential difference here will be 5.1 into 10 days to five ft to find out the work done, work done will be charge times the changing potential. So that will be 10 to part -9 times 5.1 into 10 to 5 juice. And if we simplify this, we get okay. This as 5.1 times 10 race to- for Jews. Or we can see this as 0.51 million jobs. So this .51 million Jews is the work done to bring 190'clock charge from infinity to the center at.

Question 34 states that four point charges are located at the corners of a square with sides of length. A two of the charges air plus Q and two or minus que so I put plus Q and minus que Find the magnitude and direction of the Net electric force exerted on a charge, plus Q uh, capital que located at the center of the square for each of the following two arrangements of charge, Part A is shown in the diagram here. The trend is alternate in Sign plus Q minus. Keep plus Q minus. Q. As you go around the square and company be after that. But this is the scenario they're dealing now. So part of this positive capital Q is in the center in each of them. Smaller, accused, positive or negative around the edges. Just it's clear, actually, would this may. As a legend, I think, Ah, yes, make it really big, but be labeled each of these square charges that we have Yes, there, you we'll shrink it down and his label because we're gonna rearrange the jaggery a little bit, told me. Clearances negative. Que. This is capital big Q. And this is plus little cute. So the first arrangement we around this square at such we want to find the electric, um, country course exerted on positive. Cute. So by doing that which look at the directions of each component here. So in this arrangement, if you look at this positive Cuba push that little blue que of course they black you down. Similarly, this bottom kill Put it this way this red wanna put towards itself And this red with a Pulitzer to someone's always attractive If we know in this arrangements every component, every vector component has an equal opposite vector component on the other side of the black turns cap O que That means because directors everything years the same magnitude of change of the net force in the scenario would be zero eight equal opposite vectors produces. Um uh, scenario Are all the electric courses cancelled out acting on that center Black charge. Now we move our arrangement such that you have to queue on to positive cues on top into negative cues on bottom, such as this one. Capital big Q is still in the middle. Our square still has size of like me get a little bit different scenario, but because it's positive, it'll the top left change black before exactly force on it in this direction as well as this charge building the same downward and seem but opposite direction in the crate on the side. It means each of these two there are two vectors, two vectors because there again equal magnitude of charge, they affect the same way pulling down. But when the scenario now we have one doctor from the top left charge one vector for the bottom, right charge top right and bottom left because of the same magnitude. The X component of each of these will cancel out, and we're left with oy 1,000,000 to consider the why component of each of these sectors. And, you know, again, based on symmetry, we could do a little bit more simplification because everything's pointing in the same direction downward. If we just get one, um, magnitude in the y component of the vector downward, multiply that by four to get the same effect as having four different forces pulling it down again. We don't need to get their ex meal and you to consider why. So you only need to look at one electric force component. Multiply that four, and we should. That should give us your final answer so we can do that here, so F net. Just a little bit of math represent that it's obviously enforce of Q on one plus force between Q and to Q three and Q four. I guess the generous represents one of the charges around, um, Square, but I can represent as four times F Q one. Let's say in which 1 may picking policy. It doesn't matter. So look at the one component going down to the right. Let's say so, Um, one of the ones pulling towards the bottom right? Negative church in that scenario on this angle here is 45 degrees. The length of this branch is a over to link of this one is a or two as well. That means our distance from the big Q to lower case negative. Q. On the right hand side is book are like that. That's that's R squared equals hey, squared over four plus K spread over four square both sides between you, which means a squared term. We'll say this is Ford's. Well, make it clear they spread over four as well. Which results in a squared over two is our our square component. So the Net forces of the K Q. And the magnitude of any one of the charges because they're on the same thank you over a distance term a squared over four and again dealing with the y component of this. So the y component this vector would be the coastline of 45 co sign 45 degrees multiply by four again. This is This is all one example of the force by acting on the Positive cube positive capital que charge from Altoid before we have the total force. So now all we need is a little bit of math again. A washing specify tooth is the negative. Why have direction but a little bleeding over your service? That so Just a little bit of math here we have. Um, also, this is a two of the denominator. Hey, scored over to me. Figure that out. So you have, um, to okay, little cubic. You over a squared coastline 45 was equivalent to root two over to substitute that in over two times for in the negative y direction. Yeah, and so against the question does ask for the magnitude. Amanda in the direction we know the direction is a native Wyatt direction. So, apparently to consider the Net force in terms of magnitude, you don't take a new direction. Looked a little bit of math being. Figure out that the solution is a plea for route to thank you. Q. Over a squared and again, it's occurs in the negative direction. Thank you. So this question isn't a lot of symmetry and mold of it. In the first part, read four charges Each vector cancels each other's out based on their position Kitty corner opposite corners of the square when they moved in. The arrangement with positives on top of ending is your bottom begins again, Um, certification of the geometry to recognize that we're going to compute the value of one vector and multiply by four. Recognizing the only the y component has the extra borders cancel out. Being are potentially very, very complicated. Equation is solved massively, essentially one term, and that's why buy for their, you know


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5 answers
Jozok TESU PE_ 0735OTE5 TU czthy Uures0785OTs
Jozok TESU PE_ 0735 OTE5 TU czthy Uures 0785 OTs...

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