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412q1Three negative charges are arranged as shown: The charge 91 is 0.86pC and is at distance 0.8m from charge 92 Spc: The charge 93 is 2.58uC and is at distance .8...

Question

412q1Three negative charges are arranged as shown: The charge 91 is 0.86pC and is at distance 0.8m from charge 92 Spc: The charge 93 is 2.58uC and is at distance .86m from charge 92: The magnitude of the force on 42 due charge 91 is F21. The magnitude of the force on 92 due to charge 93 is F23. What is the ratio F21/ F23 ? 1.89196875 1.6216875 2.07215625 1.4415 1.801875

412 q1 Three negative charges are arranged as shown: The charge 91 is 0.86pC and is at distance 0.8m from charge 92 Spc: The charge 93 is 2.58uC and is at distance .86m from charge 92: The magnitude of the force on 42 due charge 91 is F21. The magnitude of the force on 92 due to charge 93 is F23. What is the ratio F21/ F23 ? 1.89196875 1.6216875 2.07215625 1.4415 1.801875



Answers

The point charges in Figure $19-33$ have the following values: $q_{1}=+2.1 \mu \mathrm{C}, q_{2}=+6.3 \mu \mathrm{C}, q_{3}=-0.89 \mu \mathrm{C}$ (a) Given that the distance $d$ in Figure $19-33$ is $4.35 \mathrm{cm},$ find the direction and magnitude of the net electrostatic force exerted on the point charge $q_{1}$ (b) How would your answers to part (a) change if the distance $d$ were doubled? Explain.

In this question we have to charge particles you wanted to to fix in place along the excesses. Okay. And then they are a CMA parts uh and then we want to enter our country. The charged particle is placed between CUBA and Q. Two. And electrostatic force onto a tree. It's taught on a graph that looks like this. Me. So at this point this is true seer they were the force Acting on the on Q. G. zero. So we want to find out the sign of charge Cuban. And uh we show of Q. Two to Q. One. So to find out the sigh of um If I had the sign of Q. one. Okay. First we know that museum mm the electrostatic force on Q. 30 And then when you tree is moved closer to Q. One, the forces positive meaning the net force is pointing to the right. Okay, so this is uh that's far as pointing to the right. Even it is negative. It means that forces pointing to the left. So when Thank you. Chee is uh Okay. So for X. That's in 2cm. Okay. The nets force is to the right. Yeah. Then for eggs Greater than two cm, the net force is to die. That's all right. So then which? And then when Q. Tree is both closer. 2 to 1. Uh huh. F one increases. Okay. And F. Two decreases because of columns long. Okay. And columns law says that the the Electoral study force yes proportionate to accuse to the product of Q. Like you cues more to divide by our square. So all right. So when you move closer to Cuba one increases and after decreases. And the net force uh Ethnics is to the right. Which means that This means that F. one is pointing to the right. Yeah. Two is pointing to the left. Yeah. When I say that one is a force you to charge Q one. F two means that forced you to Q. Two. Okay. So for F one two. Mhm pointing to the right with Q. Tree being positively charged. Okay. You one must be was a different charge. Okay. So this is a reasoning why Cuban is positively charged me. And in part B you want to find a ratio of Q. Two to Q. One? Okay. So uh at exodus two cm. Okay you have at 192 equals to two. Okay. The man to have everyone equal so many to f to so many to have F1 is using customs law. Mhm. One is the go to K. Q one Q three given by two cm square. And then F two is mm You Want AQ two Q 3. You're not by 60 m square. So the name two by the same. So you can put them together so you can cancel the K. Okay? And the Q Key can also be canceled. So you to give up by Q1 is 86 cm, divide by 2cm Square, just three square, which is night. Okay? So the ratio of Q 2 to Cuba is nice and that's all for this question.

This is ejector 21. Problem number 16. So we have four point charges that were placed in the corners off a square. Um, so you can see that Let's call this Cuban first charge is negative. Then here comes the second charges. The positive off, 1st 1 positive and negative. So, um, instead of dealing with numbers, first of all, um, let me, um said off the problem in a way that initially we don't have to deal with the numerical values. Okay, let's assume that you want equals negative. Kill, right? And cute two equals Q and cuter equals positive cue again. Right? And then a few four equals negative. Cute. And it's cold the length of each side of the square to be a and were asked the net force on each charge and the direction of this force so definite for each charge in the direction off this neck. Oh, f connect. Right. So this is actually a very long problem. I'm gonna salt a little, sold it for one charge, and I'm gonna go because of how you can calculate the intellect. And then I'm going to reach a general conclusion so that you can build up all that now. So let's start with calculating the Net force acting on the first charge that I re labeled as you want. Okay, so Q one oh is going to experience an attraction as far Askew two is concerned, right, Because kids is a positive charge. So the direction all the force that after two pardon me to observe on one. Let's go at to one right this f too long. Let's write it in the vector form, you can see that it's gonna be easy. Reefer. Dussel equals to the magnitude of this force. Que que swear? Right? We're finding the magnitude over. This is between the two is a square times the unit vector you defector, Since this force is pointing to you too, while conventionally again, any force appointing in the positive extraction is defined by data. Better is if I had to correct. So if do on Vector can be written as Jake you do, or a spirit I had because if 21 is in the positive extraction now which using the same logic with strapped, figure out their direction or the vector form off u x three one now have 31 between your calculating force exerted by the third charge on the 1st 1 on the first date, right? So let's try to try to determine direction of this force, Since, um, kill three is positive and you want is negative. There's an attraction going on between the two, right? So then it's the Cuban is gonna be attracted towards Q three. Therefore, direction up 31 is going to be down. So let's remember conventionally down means negative wire direction. So Rigoni's negative J hat as far as you know, veterans concert. And as the magnitude is still kay, he was squared over a square right, because the magnitude of the 1st 1st time's managing to the second charge divided by the square of the This is between the two. Since the distance between the two is a let's put the Inter victor, there is negative J hat. So far, so good. Now we have the fourth charge. That's a serial force exerted by the port. Charge on the first charge is that for one, this F or one is it has components in both X and y directions. Okay, so let me show the direction of it. It's a negative charge. And you? What is a negative charge? There is a repulsion going on between the two. Between. Ski one is an experience of force that is away from this Q four. So this is direction of F or one Now. I drove it at an arbitrary angle from geometry and figure out what that angle is. First of all, the distance between the two happens to be. The high pot news right of a triangle has the length of a start from here, a square plus a squared square that the sward of two is D. This is between you want into four second. We've we have a right triangle like this whose legs are actually equal to each other. Then the angle here has to be 45 degrees right? Which means the angle here is also 45 degrees is you can see, for one is applied with the angle. Let me draw clearly. Wreck This is ask for one. You said that this table is 45 degrees, right? So it was good here, 45 degrees. Um, now let's train decompose. Let's find the y comported F four. Why is at four y times site off free Climb. Then what? The ex cop for it for one ex then equals two f for one whole side. 45. Right now. What is that for? One is the magnitude which happens to be king. You swear over this time let's forget the distance between these two. Georges, it's a and spirit of two. It's not a war. So a spirit of two. They were. You swear this, right? This this square This is the magnitude. For what? Now, if we're right, the, uh, Specter formal for one that that would be f for one times. First sign. 45. Right? Because this is the ex compartment times. Remember? F for one call sign 45 is in the negative extraction. So we're gonna put a negative. I have here Waas want sign 45. It's in the positive direction, though. Why comported up at 41? So therefore we're gonna put a positive g head there, So let's plug it in for 41 k squared off over to a square, Right? Basic lose to a square times too. So and then course I'm 45. What is course I am 45 is one where we're squared up to. So let's book that in there too long. Over spring. It's true negative by hat, Plus Jake, who spurred over to a square and +41 is the same again. Signed 45 is also Cypress From Equals. Both signed 45. That equals one skirt of to J F. Now let's write a little nicer. Um, que que squared over to a skirt of to a squared Negative. I had plus Que que squaring off over to a spirit of to a squared J left. All right, this is half Or want this response to force that the fourth charges exerting on the 1st 1 to die? I've been one all right. No, there are three forces. If you want the 3141 that are acting on to want that you want to find the net force acting on human, we have to add them. Oh, look at this guy. This guy to this guy, Mr Little New picture. So F net. Let's bullet for the first charge. Okay? In vector for equals two. Well, which one? We started with F to one if 21 plus f 31 What's that for one. Now, remember F to one was cake. You squared over a sprig. I had the huse were permanently square. Try that. Plus if 31 the three ones kick your skirt over our spread. Negative. Do you have excuse for her release were negative. J hat. Now we have a 41 that for one. The cake you skirt over to spread of to a spurt. Negative. I had Hey, Huse word over to spurt to a square. Negative I had for the white component you have pretty much the same. We've just grand over to a spirit of two faced great positive J hat. Right now let's group the ex components and wife components altogether. So cake you skirt over Eastbourne, Linus, because there's a minus your cake. You squared over two. Spirited to ace where I had plus Okay, juice. We're littler too screwed up to be squared. Linus, cake use where? Hey, squared J hacked. Now this term is doing iwas the X component off. Definite right. So this is six component of ever met one And this is why component of if connect one lives. Try to catch up with them individually before we go any further, So let's drop. Um, let's do this. Ash meant one x The X component of f Met equals to, um if we take this term. Right, So in both terms, in Expo in the exploded, we have cake, you square, so I'll never try and rearrange this term. He que scared over Easter minus Caju squared over to skirt to a squared. Right? All right, Now we're gonna quit the denominator so that we can do work that led to a skirt of two cake. You square over to a spirit of to, um, a squared. Alright, Linus, the acute squared over two spirited through base. From here, we have to a spirit of two minus one. Yeah, Q squared over to spirit of to be spared. Now it's time to calculate. All right, so two spirit of two minus one. Que is love. 8.99 times. Center for nine. Q is given to us. Remember, the problem is 4.1 5,000,000 belong certainly has been brought back to full square, divided by two spirit of two A. This is the length off the side of the square is 0.4 meters So we're going to square this from here. We're gonna find the X component of the net force acting on the first charge to be one time center, bro. Seven mutants do in Crete in much the same back for net one for the wife compartment. This time we have a few square might escape this word to skirt of two over. He was heard of to a squared. Right, So you few square it over to a spirit of to a square. One light is too scared of to start. Let's catch it. What it is 8.9 time sensitive power nine fines. 4.15 10 cents per negative. Three square sober to spirit of 2.1 meter square times one minus two. Spirit of two. So the net The Net Force acting on the first charge. The eight white component of it is the negative of the ex cop from here. All right, well, we're almost there now then, if we found the eggs and my components than what is ethnic acting on one and magnitude really is square. Have met one x swear lost their that one. Why square friend so soon equal to one time center here on seventh squared. Plus thinking 1 1% to 7 swears all I know it's gonna be equal to that force. Acting on the first charge is gonna be full to 1.42 times. Thanks for power seven Newton's And how are we gonna catch the direction? Always tension function tens and tater equals two f inept. Want why Divided by one X is, you know, it's a magnet. What is negative? One center bursts. Every ethnic X is a one time center of seven something here we have negative one is the ratio. So the inverse off tension off negative one equals our data data. That is gonna be negative. 45. So let's remember what this negative 45 means. This is Q on Q Jew Q three before negative 41 needs that we have the angle that is actually pointing the center. If this is the center off the square, right, negative 45 means this is 45 negative. Remember, the position is always from the positive X. So then, actually, this net force on force is towards the central, the square. Okay, So, basically, by doing the same process for the second charge, you can show that the magnitude mitt off net force acting on cue to is still one point for two times center power seven. This is true for all come so for a while charges if it actually equals the one playing for two times sent a girl of seven years. News and direction is always for he charge is towards the center. Help us square.

In this problem on the topic off electric charge an electric field were given a square off side L has shown in the diagram and in each corner off the square. We have charges point charges of magnitude Q to Q three Q and four Q has shown we went to first determine the force on the charge to queue and next to force on charge three Q. Due to the other three charges. Now, if we take the lower left hand corner of the square to be the original, the coordinates each charge will have a horizontal force on it. You to one charge, a vertical force in it, you to one charge and a diagonal force in it. You to one charge. So we have to find the components of each force at the components and find the magnitude off the net force and then the direction off this net force. And then we will draw these forces on the two charges in this diagram. So, firstly, on the charged to queue, let's look at the horizontal forces acting on this charge, so f took you X is equal to be constant K times tu que times Q over, I'll squared. So that's horizontal. Due to the charge. Q Plus que into two Q times for Q over to l squared times, of course, sign or 45 degrees. So that force will be acting Bagnall, and we found the horizontal combined off the force, so this is equal to K Times. Q. Squared over l squared into two plus two squared or two. So this becomes for 0.8284 Okay, Q squared over l squared. Now we'll find the some off the vertical forces along the Y direction, acting on the force on the charge to queue. And that's K Times two Q times three Q. This is due to the three Q charge over L squared Les que Times two q times. Four Q. So this is the vertical component off the force exerted due to the four Q charge over to l squared times the sign off 45 degrees. So this becomes okay. Q squared over l squared into six plus to square root to, and this becomes eight 0.8284 Okay, Q squared over l squared. So therefore, the magnitude of the resultant force on the to charge. F two Q is equal to the square root of these two components. If two Q x squared. Bless f tu que why squared? And if we calculate this, this becomes 10 0.1. Okay, Hugh squared over l squared. So that's the magnitude off the force acting on the took charge. But we still need the direction. And so Dita Tu que is equal to the ark 10 off the y over the X component of the forces. If tu que y over f two q X. And so it's the Ark 10 off 8.828 4/4 0.8284 which gives us 61 degrees. So we have the magnitude and the direction off the force, acting on the Tu que charge by all other charges now in Part B, we want to do the same, but for the sake you charge. So first, let's look at the horizontal component off the resultant force, so three F three Q X is equal to K times Thank you times for Q over l squared. That's due to the folks you charge, plus que into three Q times. Q over to. I'll squared Co sign 45 degrees. That's due to the charge. Q. And combining this, we get K Q squared over l squared into 12 last 3/4 times square root off to And so if we calculate this, we get the results and X component off the force. Acting on the Greek charge to be 13 on 0607 Okay, Q squared over l squared now for the vertical component of this force F three q y. This is equal to minus. K. Times three Q. Times. Two Q. This is due to the two Q charge divided by Al Squared. Minus okay, Times three Q times. Two q streak you times que rather over to l squared times a sign of 45 degrees. So this is the Y component off the force due to charge. Q. Acting on the Greek charge. So if we simplify this, this becomes minus k. Q. Squared over l squared into six last three over four times the square root off to and so calculating. This becomes minus seven 0.607 Okay, Q squared over elsewhere. So therefore, the magnitude of the resultant force acting on the charge of trick is equal to the square root off s thank you X squared. Bless F the Q y squared. So if we substitute our values from above into this equation, we get this to be 14 0.8. Que q squared over elsewhere. And so all that's left now is to find the direction off this force on the Greek charge. And so this angle Dita three Q is equal to the Ark 10 off the Y over the X component if thank you y over f the X, which gives us the angle off 332 degrees. So we found the magnitude and addiction off the force, acting on the two Q charge by all three other charges and the charge by all three other charges, and we can draw them as follows. So on the took you charge, we have result in force f tu que and the Greek charge. We have result in force if the queue

Here there are two positive charges placed along the X axis at two different positions and were asked, where would you place a third charge Q three such that the force on it, electrical force would be zero. There are a couple of things are going to do to simplify this question. Um The first thing that I'm going to do is I'm going to shift the a coordinate system over so that Q one is sitting at the origin and that will simplify some of the algebra that it's going to wind up. So if Q one is at the origin then my cue to is at the coordinates 16.9. That's going to help me see the distances and positions a little bit better. The next thing I'm going to do is I'm going to think about the force electrical from columns law. It's K. Times the product of two charges. Their absolute values over their distant squares with the direction being along the line joining the two charges. And I'm going to divide out um the Q three. So let me write one of those as Q three. Now this is developing the idea of electric field, but because Q three is going to be common to both forces, we really do not have to consider it in our calculation because if the forces hero dividing by Q three is going to still give us a force equal to zero. Um and this is the idea of the electric field that you can imagine a charge placed somewhere and the interactions happening on it without actually putting the charge into your equation. Okay, with those simplifying assumptions, now I want to think about uh where in where to put where is um the some of the electric fields. So those are just going to add up with the Q three divided out. Where is that equal to zero? And I'm going to have to think a little bit about what region of space that's likely to occur in. Yeah, happened. So let me redraw my access one more time just for fun. Yeah, I don't know if I need to make that red. Let's see. Mhm. One can get to interested in colors here, but here we go. So that there are three different regions that I could place that charge. We could put it to the left of Q one. Call that region one. We could put it in between the two charges. Call that region too. Or we could put it to the right of Q two in region three. And if I look at the forces, the electric fields, if you will um in those three regions on the left, we're going to have force one pointing to the left as well as Q two. And there's no way that that those two forces can add to zero and the same will happen in region three. And you don't even have to draw those to scale. So fortunately in Region two you can get a repulsion from both of the charges and those two repulsion will be in opposite directions. So it's region too. That's going to to work. So I'm going to imagine um somewhere in there putting the charge Q three at some value X. Now you can see why I went to one of the origin so X has to be positive. Um And less than the full distance 16.9 meters. Okay. And now what I need to do is set the magnitudes of those two forces equal to each other and we can divide out the Q three. It's a simple enough thing to do. And so we have to find the X. For which que que tu over its coordinate. Um Let's see. Okay. Yeah, yeah its coordinate 16.9 minus X squared is equal to K. Q one over X squared. And that is going to produce a quadratic for ex so it does help to pay to think a little bit about whether X is positive, negative and what value limits it should experience. And now the case cancel, we could do a little bit of cross multiplication or invert the equations if we want to think about doing that and even the micro cancels in the micro Coolum. Mhm. Yeah, let's go ahead and put 16.9 in there and it'll take a little bit of expansion of that by no meal. A little bit of cross multiplying. And I'll show you the final quadratic equation that we wind up with after some algebra. I always worry about how much algebra to include, but if I at least show a couple of intermediate steps, I don't have to get too tedious with all the crazy stuff in between. Um let's see. So that's not even, I usually like to get my leading X squared term to have a coefficient of one. Just makes life so much prettier. Okay. And then we can do our quadratic route finder. Um so look up the quadratic equation if you are not familiar with it and I'm going to need the positive route, the negative route is non buddha not allowed. It is not in the right region, I should say. That's always a reason to exclude something with a quadratic equation and I get X is equal to 7.52 m. And here is where I'd like to remember that I shifted the coordinate system so now I want to kind of shift it back so I put Q one at the origin and we are 7.52 m from that. Um And in our original coordinate system, we just want to shift it back, Bye 7.52 minus 4.7 m or 2.82 eaters. So we have our position, here's Q one at minus 4.7 m, there's the zero somewhere and our position is at 2.82 meters.


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