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Two small nonconducting spheres have a total charge of 94.6μC.When placed 1.04 m apart, the force each exerts on the other is11.0 N and is repulsive. What isthe ...

Question

Two small nonconducting spheres have a total charge of 94.6μC.When placed 1.04 m apart, the force each exerts on the other is11.0 N and is repulsive. What isthe larger charge on one of the spheres?What is the (smaller) charge on the other sphere?What if the force were attractive? What is the larger charge onone of the spheres?This problem has been incorrectly answered already.What is the smaller charge on the other sphere?

Two small nonconducting spheres have a total charge of 94.6μC. When placed 1.04 m apart, the force each exerts on the other is 11.0 N and is repulsive. What is the larger charge on one of the spheres? What is the (smaller) charge on the other sphere? What if the force were attractive? What is the larger charge on one of the spheres? This problem has been incorrectly answered already. What is the smaller charge on the other sphere?



Answers

(III) Two small nonconducting spheres have a total charge of 90.0$\mu \mathrm{C}$ (a) When placed 1.06 $\mathrm{m}$ apart, the force each exerts on the other is 12.0 $\mathrm{N}$ and is repulsive.
What is the charge on each? (b) What if the force were attractive?

This is Chapter 21. Problem number 14. We have two small non conducting spears we give indeed, shrimp. Oh, the's of the charges of these two spears 90 micro cool. Oh, so look for that to cool off his talents, right? 90 Tax center for negative six. Fulham's It's gonna be Now I thought they were asked if the separation distance between these two charges is one from long six meters and the magnitude of the force is to cover mittens and it is repulsive. That's an important part. What our cue on and cute too so repulsive means keep wanting you to. They have the same sign, right? Either both of them are positive or negative. So let's remember the coolers lock you to one of the two are square. Let's go separate unknown unknowns on one side of the equation and the nose on the other side. 221 times cuter is gonna be able to after our squares over K using the clothes law. And if you look in the numbers 12 mutants times 1.16 meters square over 8.99 times 8.99 10 to 9. Create New Year's where force where you get 1.8 months. Sensible or negative? Nine. Product. Don't you want a cute too? Now, we also know Q one plus que tu, right? So we have two equations into unknowns, and we should be able to solve forward. Right? Killer. Include two is nine times tangible. Negative six close one times. Q T is one for the 10 cents the power of negative nine. And like I said, this is a Well, they're coupe purple equations here. So there are two anyways to Seoul. This eventually you're gonna get to 1/4 of the equation. There's no escape from that. So, um well, what I would do, I would isolate one of the unknowns from one equation imploded in the other. And it's exactly what I'm gonna do here. So, um, I am going to write down to one, for instance, using the first equation, keep what he calls the 90 times. Thank you for negative six minus Q to write. And then his love this in the second equations are 90 times tempted Broom negative. Six prime askew. Two times. Que tu is gonna give me one print a central part of never night Redistribute this so 90 times come to Birmingham. 90 6 2 1 90 6 to 2. I'll input the purchases. Here. Here. Okay. Minus Q two square explains one trading texts regarding nine. Get recognized. This this is a quadratic equation of the second order of the first order. Um, so But let me put it in a arrange it in a way that we can recognize that it's a quitter. The equation. So if we take a cue to through the other side of changes, sign right and also the question okay to also changes science. It's the 90 negative of mine. Intense Enter burning six few, two plus 1.8 tense There for a minute is not equal zero. Now, you can see easily Right to credit equation A into credited formula. If you remember, there are two solutions to these type of of equations. Um, you want Q two Assuming that we have to look, we're gonna have to solutions right. Negative B coefficient, B plus and minus strike that their corresponding to separate solutions be spared months for a C four times. He's a c can see over to a Now let's see what a is here. In our case, it's what B is negative. 90 times after coordinated six in C is born from a text after Perfect man. Now then, if we apply the credit typically from una que tu equals negative be right. Negative B negative. Negative excuses. Positive. So this is 90 times center covering six plus. Let's take the plus. You're going to see that you get the same answer even if you take that negative. Um, in them we have spare root of B squared. So negative 90 times 90 10 to 4 minutes. Six squared minus four times one is one times C 1.8 tanks, tripper Negative. Nine over two times. One his two. So from here, Q two is gonna be found to be 16 times 10 to the Baron, maybe of six. Who else? So now, since we know Q true, we can easily excuse the first equation in order to be able to calculate the well. Actually, it's better here, right? We had isolated this Q one. Then it goes to 90 times 1 90 9 to 5 negative six minus 60 times. Thanks for major six. So it is 30 times sentry broke. It's like school is you can see they have a seat inside. Really? That's good. Now, import fee of the problem. We're even this time, the force as an attractive force. That's the only difference in party. So now the magnitude is that everything is the same on Lee, that the force is attractive, Which means they have one clue to have opposite sides. Right? So only changed it. Rig. Unlike brick, I'm gonna follow pretty much the same recipe. The only difference now, because the force is negative. Remember, our 21 times two equals f our sperm over. Okay, this time it's not gonna be equal to something positive. It's gonna be because of the forced maker to be negative because they have opposite sides. I gotta walk in any time, Senator. Former negative nine is What we're gonna have is a is you want to ask you to still be long. Plus, que tu is the same lighting kind center. Probability of sixfold spread. So we're gonna have a little bit of the same recipe than, um, isolate. Let's take you on out of here too. On equals 95 cent. People made of six months. You too, and then put it in the first equation. so 90 times six minus Q. Two times Q two equals negative 1.8 times sent to her native nine again 90 tanks that for nearly 2 1.8 9 90 6 to 2 minus you two squared equals negative 18 times that burning mine again. Reed, you're facing a quadratic equation. So if you two squared, I'm learning this 90 test after her negative six q two minus 1.8. That's the difference. Time spent. Two remaining nine equals zero. So now, again, a is the same. They is more. He is safe. Negative 90 times 1.8 2 9 0 90 10 to 26. But see is now negative, and that's gonna change for answers. But again, using the quadratic equation then due to is gonna be equal to 90 times center for maybe six. Just use the positive value square root of game. They get a 90 pound center birth. They get a six squared minus four negative one point in time. Sentry Ever leaving this time, right? Spirit goes everywhere for burn from fear. Que tu is gonna be one of seven hand central needed six forms and again he will on people's 90 times. Some people make it six minus 107 times something burning of six males. Thank you. One is gonna be needing a 17 times went to Birmingham. Six schools is Casey. The signs are Oh, that's it. So everything was great between the minus. It'll the off the force is negative, which means the there is an attraction boredom.

This question we have to uh small charges, charged spheres. And then uh Each have 2, 1 and Q2. And we are given that If you want to ask you to is equal to 15 Micro column. And then there are space And two m apart. The force on them is when you turn uh in one year 10-K. And they repel each other. So we want to find uh charge on the sphere with the smaller charge. So to solve this problem, a using who runs slower, see F goes to king. He wants you to buy buy our square. Okay? So you have mm 21 times 50 times 10 to the negative six minus uh to one. Okay, If I buy our square is equal to one. Okay? So uh so we expand uh the things with Cuban and this is equal to a squared divided by K. This is two squared by by 8.99 times 10 to the nine complete days. You get 4.45 times 10 to the -10. So you re arrange you get a quadratic equation to one square -50 times 10 to the -6. You won plus 4.45. It's 10 to the negative 10 goes to zero Kenny. So using the quadratic formula. Okay. So Q one is oh uh many speakers minus B squared minus four A C. Square roots. So Then there's divide by two. So you get two answers. Q one is to go to 38.4 Micro Cologne or 11.6. My curriculum. Okay. So. Uh huh. Charge on this here? Yes. Uh smaller charge. Yeah. Is it good to 11.6 Micro column? Okay. But if you want it into SF will be charged micro club. Well, so yeah, the answer's for this question. Okay, and that's all for this question.

For this problem. On the topic off Coghlan's law, we're told that two small non conducting spheres have a total charge of 19 micro columns when the two spheres are placed a given distance apart. Then we have given the force that each exerts on each other, and we're told that this force is repulsive. We want to calculate the charge on each sphere. We didn't want to redo the calculation if the force had to now become attractive. So if he forces repulsive than charges must be positive, since the total charge is positive. So if we call the total charge Q. In Q one plus Q two is equal to the total. Judge Que and we know that the force between the two charges is K Q one que two over the distance separating them squared B squared, and we can write this as que que into q one multiplied by Q two, which is Q minus Q one all over the square. So this implies that with some rearranging, we get Q one squared minus Q Times Q one plus yes, D squared over K is equal to zero, so essentially we have a quadratic equation in Cuba. So if we use the quadratic formula and we solve this political creation, we get Q one to be equal to que class of minus the square root off Q squared minus four times f d squared off. Okay, and this is all divided by two. So now we can substitute our values in here, and we confined this value for Q one. So that's a half. And since all our values and S I units will suppress the units here and this is 90 times 10 to the minus six and that's cool arms plus or minus the square root. Yeah, off 90 I think times 10 to the minus six minus four multiplied Bye before 12 Newton's kindly separation systems between the sphere zero point to 8 m that squared divided by que, which is eight point 98 eight times 10 to the nine Newton. Me too squared Coolum Square So well, just suppress the units we have s i units for all our values. So if we calculate this, we therefore get a value for Q one and Q two. So we get Q one to be 88 0.8 times 10 to the minus six columns. And so, obviously, from there we can see that Q two is 19 ministers value, and the second charge as magnitude one went to times 10 to the minus six cool hours. So if the force between the two spheres is repulsive, these other two charges on each sphere. Now what happens if this force was attractive? But if the force is attractive than the charges off opposite Sign and the value used for F must then be negative. And other than that, we use the same method as above. So again, we write Q one as que glass a minus the square root off Q squared minus four times F D squared over K, all divided by two. And as we did before, we'll put in our values and suppress the units. So that's a half into 90 times 10 to the minus six columns, plus or minus the square root off Q squared, which is 19 times 10 to the minus six square, so that as well in the first expression is 90 minus 90 10% to the minus six old squared, and this is minus soul. Another force is minus 12. Newton's on the separation distance, which is still 0.28 m all squared, divided by the constant K which we know is 8. 29 88 times 10 to the nine in S I units. And so if we calculate, we get Q one to be 91.1 times 10 to the minus six columns and this is positive. So therefore the charge thank you too must be negative. And this will be minus 1.1 times 10 to the minus six Coghlan's So that's the charger needs fear if the force between them is now attractive.

So and this problem. We're told we have two identical conducting spheres, and we know they're repelling each other at a certain distance, and then we connect them, and they're gonna probably each other the different force. And we want to know what was the original charge on each, um, spear. So let's write down the givens, and then we can discuss how to do the problem. So was called F the original forests F one. So that's point no. 50 new ends in the original distances 0.25 meters. Okay. And the, um, second forest is 0.60 And we want to get the original charge on each, um, spear. So what's right down? Let's sort of take this approach. Um, let's approach this with a Newtonian perspective. So, um, Miss double track something, um, you okay? And they're identical, which I suppose is important. So if there are identical spheres, I think they must have the same charge. Initially, let me pause and check that. Oh, I see. No, I cannot be the case because, um, otherwise connecting them would would not change the charge distribution. So originally have these two chart two balls And then, um, I'll call this one A and B And then you can say that the forest between A and B that's what we know is F one. So I'll call that one. Um, that's going to be equal to the charge on a comes a charge on B times K over r squared. And so we don't know what we don't know what these two are. We have this and we have this and you have this. So what this equation gives us is the product of Q and Q B. So we can kind of keep that in mind. Um, now they distribute their charge. So, um, let's say that afterwards, um, there's, uh They redistributed their charge. And now you have a charge Que a prime. Here. Thank you. Be prime. Um, and then we know that we know the forest is equal to force to. And then that's que a prime q b prime over r squared times K. And our goal is to get the I guess there goes to get Q and Q B s. So I'm gonna just take off this cube. Doesn't matter. Um s o the main idea. We want to use is charge, um, conservation. And actually, before I dive into that, let me make one more simplification. Since Q A prime and Q B prime are equal. Let's just say that Q prime. So it's followed the charge on each sphere afterwards. Q. Prime. And so now, since it's the same charge Oh, yeah, and then it would be the same charge because once they're connected, the electron, the excess charge, will spread out as much as possible, equally distributing over each sphere. So so now this is our formula for the force. Between the two charges, Um, are the two balls given that they each have a charge? Q Prime, and then this is the original force. And now we want to use charge contribution. So we want to say that like, you know, this total amount of charge is equal to the total amount of charge here, So the total monitored originally is Q A Plus Q B. And then the total of my charge finally is to Q prime. So we have this formula, this formula and this formula. So it's kind of three unknowns, three equations so waken solve for, um so we should be able to solve for all of our unknowns, I think, um so where should we begin? Let's I'm tempted to just seven for Q prime here from from here on out. It's an algebra problem, by the way, so you don't have to follow exactly what I dio. I think it's sometimes more challenging to follow somebody else's algebra than to just do your own, so you find it difficult. Maybe just try it on your own. Um, so anyway, so I'm gonna sub in for Q prime. That's que a plus q b squared. And then we're gonna be also scoring this four. So four r squared and then you have this K. And, uh, I think our must a note on our I guess there's only one are there couldn't be multiple. Ours were. Also, I think this problem wouldn't make sense, so let's just assume our stays the same. Um, so now we have this equation in Lao. Let's sub for cute. A Let's say, um so que a from this equation is equal Teoh. But in terms of you being able summit in here, so cue a is equal. Teoh F one r squared, divided by you be times K. Um, maybe a little messy because it is squared. That's all right. Um okay, so now we'll stop this in here. Um, for Q A. So I'm gonna come over here just to continue this line. So cue A, we know is f one r squared plus Q b. It's over, Hubie Times K. And then that's plus Q b. This is looking messy into the units. I guess the units let me just check that they work out at first, word would be Okay, Well, hopefully get simply, This is are you looking a little messy? So at two is equal to this divided by four r squared. And then now we just have a nosh. The equation for Q B so we can solve it. Um, so let's bring this four r squared over so that we have four r squared up to divided by K is equal to this whole thing. And let's just take a square root while we're doing so. We're square rooting this square with this, and then we can say that's equal to F one r squared over Q b. Okay, plus cube be, um and yeah, so Let's see what's next up. I guess we can put, um, these under one denominator. So this I need to multiply by Q b k to keep it the original Q B. And now we can move the by both sides by q b. K. So then you have Q b k. Wow, what a mess this problem turned out to be. I really don't see any other way, But I am a little surprised It all the algebra All right, So bring not over there than f one r squared plus k q b squared. I'm just, like, really curious if the units work out here. So Forest Times this I guess that is gonna be cake, you square. Yeah, these two definitely have the same units. All right, I was really hoping something would cancel. Um, and we can bring this to the other side and we'll have a quadratic that we could just put in the quadratic formula. So f one r squared plus okay, Q B minus. Que que he be que screw of four r squared up to over. Okay, Um, okay, so now we have the quadratic formula. This is quadratic and Hubie. Oh, Yeah. Oops! And forgot this Q b squared. So those are these air are cowfish ships, books. This whole thing is a coefficient. So then we will get the Q b equals, Um, so we'll do negative Mrs are beat like for quadratic is a X squared plus B x plus c and then exes are Q b, you know, then B is gonna be the this coefficient. So, um so negative. This Yeah, I'm just using the quadratic formula. If it's not clear, so negative b, there's gonna be plus or minus the square root of B squared. So this thing squared so K squared for R squared F two over K. It's a little simplification minus four times a is K and then see is up one r squared. And then this whole thing is divided by two times a and in a, it's just K. Okay, let's simplify this a bit. So those K cancels with this K and, um Oh, that's nice. Then you can, um, then you just have to minus You can take out a four k. So let's go ahead and do that. Oops. And then why did I write Cavey here? So I should be negative BSO. It's just negative. Okay. Negative of this. Which at which I wrote, um And then, actually, So Kate of the first divided by Kato, the 1/2 is, uh, Kate of the 1/2 so that I can take this and simplify it and put a k here. Um, yeah, I'm to know yesterday and algebra. And now I'm seeing why it's so important to be able to do these manipulations. Um, some of these problems could be really thorny when it comes to doing them. Um, for R squared up to k or actually weaken. Yeah, whatever. I'm just gonna leave that. I think of doing too many simplifications. Might get confusing. Um, And then so the Adami of the Square root And then week again, we can take out a four r squared k from both terms, and then we have up to minus up one, and then that's divided by two K. Well, this from is really getting long. Um, yeah, I guess we can just Well, we can take out a k from both and then bring it down here and then make that square rooted square rooted K. And then we can take out, take the are and put it on the outside. Um, and that makes it are the first power. And then so we can multiply this whole thing by our and then I honest to I guess you can also dio So we have a two are No, This is really not that bad. I like it when these problems, like, really clean up. Um all right. And then this to cancels with this, too. So I'm cancelling. I'm just gonna castle it. So goodbye. This too good by this to and yeah, now we can plug everything into a calculator, so I pull out my calculator, so I want to do negative square root of two with prank. I go 60 Um, And then let's do the plus, um, first, plus the square root of f to myself one. So that's, um, 0.1 Now for that one. Yeah, that's why no one. All right, point of one. And then we want to take that whole paying in parentheses and multiply it by our It was just point to five, and then we wanted to buy that by the square root of of, um Que? Que is nine times 10 to the nine. So let's see what I get for that. I got, um, negative. 3.8 times 10 to the minus seven. And I'm just gonna write down both solutions and then think about which one to pick after. So that was the, um, plus solution. Let's do the minus the minuses. Negative. 9.9 Right? Right. Um and this isn't cruel ums. At least I'm getting numbers that are pretty similar to another numbers I've seen. I am. So I'm not using the right number six eggs. Three point. I'm just confused situation of threes with three pointing to okay with our Q b. And then is it okay and then let's go back to this formula here. Cuba is F one. You re right. It f one r squared, divided by Q b K. Um, and so we can get Q and either case so. So if we want to do like, 05 f one times 25.25 right, divided by Q b. So negative 3.2 times 10 B minus seven times nine times 10 to the nine. And then I get negative. 9.90 nice. So you can kind of see that the two results corresponds to, um, the fact that you can have, like, you can have ah thing. And the Q b can have Think the other thing and then or you can, like, swap the properties. I feel like that did not make sense. So I'm just going to write this out until it becomes clear. Um, s O 1.9 times two minus seven cools. Yeah, So you can see. And then I'm guessing, if I calculate by plug in this for Q B, they'll get this for Q A. So then this would be This is the original pair of charges. So Q and Q B is sort of general. So then I'll say that the two possible charges are this and this, and then I suppose they could be positive as well. I'm kind of wondering why I got negative charges. If there is some assumption that I made, um, I'm not sure, but I do stand by the math that I did. So I was already getting pretty long, so I think I'm just gonna end it here


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