This is checked in 21 problem number 84 were given a profitable arrangement, and we are asked to show that the electric field, longest perpendicular by sector, is proportional to one over R to the power, for That's what we want to show at the arbitrary point. So in order to do that, let's is it's drawn here. The the electric fields from the positive point charges Q and the queue on the other side of the excesses. Let's go. These he plus and the electric field at that generated by negative too Q as e negative. So direction of the negative is gonna be towards the negative to cure, obviously. And he positive is gonna be away from because it's due to a positive charge, right? We have two of them, so this is pretty much gonna be the arrangement. So if you want to know the Net electric field, then how are we gonna write it? Let's please pay attention defect at the E. Poor positives. The ex components of these two vectors, they're gonna cancel each other out. What we're gonna have is twice of the positive of the electric field due to the positive charges. Time the sign off this angle. Because if this angle is data, then this angle has to be a favor. Right? So when that's the case, then the white component of this he puffs has to be proportional to sign there. So then, overall, in the positive direction, we have twice the electric field due to a single charge. Times sign off the same ball minus in the negative Y direction. We have B negative, meaning the electric field. Due to this negative to charge. Now let's hear what the positive is again. This is the electric. Due to a positive single charge, it's gonna make a cute over the distance from the charge itself to the point they were interested. So this is a distance you were talking about. Let's call it a D. He squared, right? So what is D in from the hot new stir up the war in there? It's gonna be equal to our story. Plus, elsewhere scare wood off this right and also over at it. Let's figure it was signed today. That would be the divisional scientist is the opposite. Our overhead hot news. D right. So are you. D is far square house, they'll skirt 1/2 our So what is E Negative is K to kill over the distance from the charge to the point average. Interested in It's our square. Right? So we can put all that information in the main equation twice the e plus Que que over instead of the square. Let's put r squared. Plus elsewhere. Scare of this is gonna be cool. To what? It's gonna eliminate the square root sign. Heda heda wien r over r squared plus Hello, squared spear. Ruutel, This right, this is our sign. Data minus que tu que over our square. Right? This is metal. So from now on, we're gonna We're gonna do a lot of eligible. Okay, So the first thing we're gonna do let's look at the common terms that we have in both of these terms. Like to okay and cue. Right. These air common terms. So then, to cake you, um, what we have is the first term to take you is are over, right. The denominators are gonna be multiplied. So then it's gonna be our spirit. Plus else Carrie, three over to write minus long over are square, right when redistribute. We're getting the previous. Now we're gonna do several tricks. What we're gonna do is we're gonna try and quit the denominators here so that we can do this. Subtraction. How are we gonna do it? By extending the first fraction by r squared by expanding the second direction. My Oscar. I'll square three true. Okay, so from here, it's good. Next beach Net Electric field is to take you. Um, now, what did we do? Expanded the first section, but our skirts are times are scared Is gonna give us our cute here. Over the denominator is gonna be our square times Our spirit times our spirit plus small spirit Power three over two for both minus, um, one tens are scared. Plus elsewhere to power three. It was very cross aspirate free, however too, right? So since now we have one denominator what we could do, we can take it out of there. So to cake, you sober are square times are spare parts all screwed. Three over two. And here we have Our cute miners are squared. Plus l spared three or two. No. Let's, um, try to take our cute out from here to take You are cute, and this is gonna be one minus. Right? So what we need to do, we're gonna regroup R squared plus squared. Then if we do, um, r squared here when we take the power of three over two, let's see what this is gonna give us our cute terms. This is gonna give us the previous line, so Okay, that's one right? And in denominated we have are spared our stiff. Plus l squared three over two. Okay, Now we're gonna do the same trick with the denominator here. Okay? To cake. You are cubes over. Someone try and let's just do this. I'm gonna multiplying to denominated by are cute and then behind it by another r cubed. Okay. Have been like what I have is our to the palace Five from this multiplication, right. And then I have are scared. Lost out, squared over are spared to the power of three over to. Okay, this term remains were saying one minus our scarecrow spells. Good morning. Are scared. Three over truth. All right, so then, to cake you are cubes over Harp's power. Five. Let's get what this term equals. Two are scared of last. There is one was false. Whereto are spared. Three over two right here, one minus one. Plus, it'll spiriting her our spears to power three over truth, right? Oh, from here, What we're gonna do is that, um we're going to start using our approximations. So the first first, let's do the binomial extension of remember, this is a different term here. Okay, so from the binomial expansion of this, so one plus health squared over R squared to power through, over to he's gonna give us one plus three over two. Elsewhere are square right extension. So whatever you see, this term, we can equate that one. Plus, the reality all scared are square. So then, if we do that the next electric field, then he's gonna be equal to Thio. More acute soldered onto the tariff. I remember individuated we have this term, so one plus three elsewhere, it over to spare. So we have one. Linus, come on. Plus three over to l squared over our squid because you can see thes wants are gonna cancel each other out, So this is gonna be able to to cake. You are cute over our two power five, um, one plus three spiritual to our squares. So we have negative three over to all spirit over to our square. Right. So now we're gonna do our of the other approximation. We had said that our was actually much, much greater than out Correct. If our is much, much greater than l Look at this turn here this term, if art is too big based on Elvis firm goes to zero. So in denominated, we only have one there. The total the net electric feel is to cake. You are accused over art. If l five times one, I'm writing it, it's a negative three over two squared square. Now these twos, we're gonna cancel each other out. We have our scared. This gonna kill two of the ours. So one R is gonna be a breeze is down to four. Then what we have is negative. Three Que que ele Spared over horror to power or right so well, that's what we wanted to prove. Anyways, that the electric field is decreasing is a function of one of our two power for and then also I want you to pay attention to this negative science which means things towards the magnetic electric. It was towards the negative awards. I get it to you to torch and he is Depending on it is a function of one arrived the barrel for.