In this question, we have ah, uh, charge disk with a non uniform surface charge density, which is six and at times are a And then, uh, we are interested to find out, uh, actually passing the question. First, we need to find an electric potential at Point X, um, which is ex a distance X away from the center of the disk. And then after that, we need to find an electric view and find out electric few and, uh, very far distance. Okay, for X is much, much larger than a Yeah. So, uh, to do part A, we are given a continuous charge distribution. So we are going to break the charge this into smaller pieces, and the smaller pieces will be small, uh, 10 rings. Okay, so I'm okay. You'll be like, thin rings. Okay, So And if you refer to the book, see the electric potential due to the ring will be Q four pipes or not hours times square of our square plus x square. But since you are looking at small little rings, it's a small lettering, but, uh, brings with infinite testimony mark charge. Uh, a cube. Okay. So, um and then we have the distances where we ask where plus x squared. So not that in this question, the access a constant We are going to integrate respect to our way. He too is our customer amount. Charge is equal to Sigma D A. Yeah, and then see my c might not. Times are over a and then the d a is two pi r p r. Yeah. So the two pi r is a circumference of the ring, and then the r is with of the ring. Okay, so, uh huh uh, circumference times a small length small, which is a small area that then ring covers. So, uh, you can put everything together, So the ring is they are not are a I'm super ir on the r. You're right by for pie. Absolutely not. Square root square, I think. Square. Yeah, And then if you want to find ah, the electric potential. Okay. You do that desk. Yeah, you'll be some ish Integrate devi ring. Okay. And okay, so from the both expression, you can see that uh, two pi and four pi can be simplified and then you can actually pull out the Constant Sigma over two X on a A and then you have great from zero to a Our square by square root are square X where we are. Yeah. So, um, we have this integral that we need to do. Family cannot do it easily, so we need to do some substitution. So if you look at the diagram, Okay, I'm going to call this anger data. Yeah, um and so if you from the diagram. Okay, find data, yes. Or or work square root by square plus x square. And then we also have tangent data, the sequel to our over X. Okay, so, yeah, this is distance X. So we have are it goes to X engine data. So I'm going to do I'm going to use substitution. All right? It goes to X tangent data. So if I do, the are you will be x times you can square data e data. And then I also need to change the limits. So when are it goes to zero? Yeah, uh, to increase zero. Yeah. And then when? All right. Good to a Hey, that would be our engine. Uh, they were X. Yeah. So I'm going. I'm just going to evaluate this, uh, integral first. Okay. Okay. So, um, so this integral from zero to a r squared, divided by the square of our square pass X where we are after the substitution, you look like Yes. And then, uh, so you have our square divided by the square of our square. Uh, class X square. So some you have one part of it as a sign. Data. Okay, so you have. And then the other parties are so the prices extension data and then signed data. And then you put the r replace that. They are your data. You have X sequence, square data. Uh, data. Okay. Okay. So, simplify. This is what you get for other X square, because they are constant. And then so tangent data is signed over design. So you get, um, you can actually get secret data and June Square. Yeah. Yeah. So second square data is one of the core science square combined in one of the science data. Get tangent, and then you have everyone second data. Okay, so this, uh, thing can be integrated using trying use a friend so far. Yeah. And you get, um, engine data. Second data minus hyperbolic. tangent. Hyperbolic tangent. Uh, science data A from zero to attention. Ah, over X. Okay. So, yeah, this is, uh, scary part, But you can do it using Wolfram Alpha, So substitute the okay, so the lower limit is zero. So when data is zero, everything is zero. So we don't need to care about the lower limit. You need to care about the upper limit. Okay. So, tangent, uh, they don't get your ex. So second data is gonna work with science data, and then you go back to the diagram here. The coastline data is equal to, um X over square root of our square plus x square and seconds later, you just to take the reciprocal. So you get a square root of a square plus x square. Because this time the R is a and then the Rubber X Okay. It's also half price. Okay? Yeah. And then, uh, apart at the bank, you just be a over square root a square plus x square. Okay, is then you multiply in the X squared over two, so you have half or a over two square root of X squared plus x square minus X squared over two. Ah, hyperbolic attention. Right. Okay. Hyperbolic tangent of a divided by the square root of X squared plus x square. Okay. Uh huh. Okay, so this is the integral, and then you put back into the electric potential. So you have someone on over two x on a You were too. Square, square, square pass, X square minus X square, too. I have a public pension. Uh, in your words were of a square. Thanks. Great. Mm. So this is the answer for heart aid? Yeah. Now you proceed to find an electric fuel. Okay, So e equals two negative DVD X. Yeah. So the constants just you mean and change, and then the inside able to, uh, you have so differentiate the square root of X squared plus X, where you're going to get to eggs. So get the half from the power. Then you are going to get You are going to reduce the power from half to buy one. So we got minus the power to the negative half, so I'm going to put that Yeah, uh, denominator. Okay, then the next time you are going to do product rule, so try to see that ex uh, have a bolic have abolished tension, uh, square root of X squared plus x square, and then you need to differentiate a hyperbolic tension. So this is, uh, one over one, minus the square over the square, past X square, multiply. And then So this is the Yeah, I need to do change room, okay? And then differentiate the inside Respect to X. You get, uh, a times minus half the myself is the power. And then you're also going to get two x from the university of of a square plus x square, and then you Then you reduce the power from minus half, too. Ministry have. So it's where? Plus X square. Mhm. Negative tree. Not negative, but this tree are mhm. He looks scary, but yes, it's actually quite good, because it can simplify it to nice tones. Leader. Okay, just keep going. He saw the two and the two cancer. So we have a X or two. Where is square pass? X square, minus X hyperbolic pensions. Yeah, We were school of the square plus X. Where he here he does. There's a minus sign here and then becomes a plus. Either two cancels you have X squared over two. Okay, so the term here, uh, I'm going to combine the fraction in the bottom. So you have one minus a square by a square plus x square. Right. So one minus a squared B but by a square Class X where this is, uh, square plus X squared, minus the square right by a square plus x square. So if you do the reciprocal, you get a square plus x square by x square, and then you have a X B bye bye, uh, square past X Where the tree house. So we can see that the X square councils or X squared plus x way. This one together half. And you see that is the same Lazar from the first time so you can actually combine them. Do you have a x rabbi square off a square classic square minus X hyperbolic tension or squared off the square pass x way, Then access a common factor. So I'm going to take it out, and at the same time, I'm going to change that sign. Okay, Like multiply the negative side inside. They are not X over to Exxon or a hyperbolic tension in a square of n squared plus X square minus a over square off E x or square Rocky square, past X square. Mm. Yeah. Okay. So this is, uh, 22 of e. Okay? Yes. Yeah. If you want a vector notation, This is how it looks like. Okay. See, now you move on to Passy. What happens when, uh, access much, Much larger than the radios. Okay. All right. So, um or X Much, much larger than a. So we have a or X Be much, much smaller than one. Okay, so, um, all right, there's no X here. Sorry. Because the access be taken up. Yeah. So, um, so far, so a or square it off. Uh, a square class X square is going to be much, much smaller than one. Yeah, because the denominator is going to be X. Okay. So, um okay, so we are going to have, uh We also need the relationship for, uh, hyperbolic tangent X in your case on small X. Yeah. Thanks. Much much. Uh, less than one gets X right, X plus no minus. Uh, no, essentially a plus. Yeah. Plus, uh, one third. Thanks, Q. All right. So, um above expression e We have e equals two sigma, not X over two x on a, um, hyperbolic tension. He over square it off. Yes, square plastic square minus a square off a square past x square. Yeah. So, uh, so the hyperbolic tension we get over square a square plastic square last one third. Oh, yeah, Cute is I buy you square pass X square with three half and then minus You will square off a square plastic square. Yeah, so this time is gone, And then we have seen a lot of X over to exile. Not a one third. I'm going to pull out X Square Insight. Okay. And then you have one plus a square or x square. The negative three half. Okay, So if you look at this, uh, expression we're here. So we have this term that is very small. Okay. And in this expansion, we only we will just take one out, because the next time will be time with a square times x square and you get into the fire divide by X to the fire, which is so small that you don't need come anymore. So, uh, we're here. You can just approximate this into single no X divide by two X l a one did. Thank you. Excuse me. Okay. Yeah. So you can't cancer. One of the things we're here. Okay, so you reach this step, uh, seem a lot, uh, a square. All right. Six. Absolutely not, uh, x squared. OK, so until now, we need to find a total charge. So because of what we expect to get, uh, electric fuel due to a point charge when When is very far away. Okay, so, yes. Have a separate step. We're here to the charge on the desk. It is is equal to, uh, sigma and the a A. This is, uh, Simona. Times are a two pi r. You are from zero to a You got all the constants in mind all the time to pay, uh, divide by a and then your leverage integral off square. We are from zero to a So you get some are not, um, two pi. Oh, a times I wanted to thank you. Okay, so you get to debt to pay over key. I think my knife you square. Yeah. And this is our cue. Yeah. So, um so check you over two pi. Is he going to Single? Earner is square. So you just replace uh, g q or what? Uh, six x or not X square. And then then you have your two pi. Yeah, and then you simplify too. Get you over for pay. Absolutely not X square. Yeah. This is what we want to get. Okay. Yeah. So this is the electric fuel. Yeah. You, too. On charge. Yeah. Which is why we expect together for X much, much larger than be So that's all for this question.