A solid non-conducting sphere Or radius R is uniformly charged with volume charge density of p_14.Find the electric field inside the sphere. I < R a. zero b. pr ...

Solid Nonconducting Sphere A solid nonconducting sphere of radius $R$ has a nonuniform charge distribution of volume charge density $\rho=\rho_{s} r / R$, where $\rho_{s}$ is a constant and $r$ is the distance from the center of the sphere. Show (a) that the total charge on the sphere is $Q=\pi \rho_{s} R^{3}$ and $(b)$ that $$ |\vec{E}|=k \frac{|Q|}{R^{4}} r^{2} $$ gives the magnitude of the electric field inside the sphere.

Days within Chapter 22. Problem 39 here. So it says a non conducting sphere radius are not is uniformly charged with charged in city row E. It is surrounded by a concentric metal conducting spirit. Kal Shell of Inter Radius are one outer radius are, too, which carries a net charge of positive. Cute. We want to determine the resulting electric field in the different regions. Okay, so let's go ahead and draw this else that we have a sphere of radius are not. And this is non conducting, uniformly charged with roadie. Outside of this, we have another's miracle shell with small radius are one big radius are too, and it has a total charge. Cute, so Q equals cute and plus que out because it's conducting so we can only have charged at the at the surface is the inner and the outer surface. Okay, part eh? Asked us. We want to figure out the fueled for or less are not will immediately. Let's figure out we gotta figure out how much charges in closed. So we drawn calc and surface, and we know that from the calcium sphere, the electric field has given us how much is charges closed over four pi. Absolutely no r squared. So cute. Enclosed for our less than or not is given by deed charged density tigers How much, boy? And we've gotten through or cool. So if we plug that into our equation right here we see that e now equals ro e r Over three slums often. Pardon me, this is for ours. Greater than are not, but less than arm one. I'm coming. So the enclosed charge in this case is just all of the charge included in the small sphere. So we drove us out here. We're not including any of the charge from conductor yet so cute and closed his road E times 4/3 pi are not cute. Plug that into our equation for the electric field and we're left with wrote e r not cubed over three Absalon not r squared. Cool. Now let's move on to part. See, So this is for our greater than our one, but less than are too. And this puts us inside of the conductor, so we don't even have to do anything. We automatically know that the electric field is zero Part D finally asks us no for our greater than our two. So now of total enclosed charge is cue from the conductor. Plus, however, much was enclosed in our small sphere, so that's the total in charge of quote charging closed. So our electric field now becomes que plus wrote e times for third. I are not huge all over. Four pi slow our school. Or, if we want to simplify this a little bit, we can right Q Over or pie. Absolute murder plus Road E r. Not cubed over three slung mud all times one over r squared. Cool. And that's it for this problem.

So we have a spear off radius. I'm not. I mean, it has ah charged density distribution. That goes as PR we want on the total charge here is Q. You want to find bee So to find the total charge, we simply know integral Tru TV of the Sylvie called Oh integral Roy's br Devi, as you know for a spear is or by r squared d r. And we degrade from zero to are not that so this is our cube we are doing is our part for before we have four maybe times are four by four from hard Dr Zito giving us Hi We are not by four, which means he is equal to Cuba. I are not fire for no for party. You want to find out the field inside this fear? So inside the sphere, let's consider a Gaussian surface off radius are no. The flux will be the integral of field times area. The hard work of people and area should be called the charge in close, divided by a salon. Now charge enclosed will simply be the integral off TV. So that's integral from zero toe are because the only considered the charge until the distance are. I'm Brody. You know that rope is beyond on TV is for pi r squared ear Excuse us you are part for by are not part for mind you that I was substituted BZ will do easy will do cue by I are not part for unsealing the fight So once we have the in close shot that's it. No, that electric bill is constant so integral Eat our dear will simply be into four pi r squared which is equal dough charging close by Excellent Not on We saw the child enclosed was Q r squared by our our part for by are not far four divided by accident not giving us electric real physical do one by four by Absalon nods you ask where by are not part for when I was listening Are by the way no outside this six space fear when you are is greater than or not you consider got in surface spherical cost in service with radius are on integral e d year is equal to charge Enclosed, derided by answer or not. So the idea is obviously eat times four by our square Now this is called a charge enclosed, divided basil or not betweens. The electric field will be you bite or lifestyle or not are scared just like a point judge. Also remember that the direction of the electric field will be radio in both the cases really outwards, depending on the sign off.

So in this question, we have the volume charge density instead of a solid sphere Is row not are over a close our row And we want the total charge for a So Q is the integral of road Devi And then DVD is, um eso is gonna be four pi r squared d are so DVD You can think of his concentric shells. Um, everyone I integrate that. That's our db. Hey, we want to multiply by row and then row. I'm gonna seven here. And I know this is like an unconventional order. So sorry for the awkwardness. Um, so this is the integral that we want to do. Let's take out the constants. Keep it simple. So we have for pie, there's a constant and then rover a row, not over a. And then we have our cubed er and then the integral of our QDR is are the fourth over four. So that's gonna cancel out this for over here? Oh, yeah. I'm in. Our is gonna go from zero. Um are what is the actual radius radius? A. So we're gonna interview from Syria? A. So then we're gonna do pie row, not a to the fourth divided by a is a cubed And, um, we want the electric field strength within this fear, um, of the function of the distance. So in general for that he you want that? You know, the integral of ee dot d a is Q and closed over Absalon not and then in general, que enclosed, We're gonna integrate. We're kind of doing the same thing Except where we don't integrate out today we only grant a great out Teoh. Um, to some are I'm just going to say we want he is a function of our prime because there's so many are so our prime is the the distance from the center of the sphere. So Q and closed. Um, so we're gonna have I'm just gonna kind of do everything starting at this step, we're gonna integrate from, um zero to our prime, and so integrated from zero to our prime gives our prime to the fourth. And then we want to divide by four. And so that simplifies. And then we get high row, not divided by a our prime to the fourth. And then you Nadia is e times the area of the Gaussian surface and so Gaussian surface would be like, This is the whole sphere and this is our problem. A radius are prime. It would be the area surface area of this fear. Spherical shells us pi r squared times our prime squared times, for I kind of regret doing our prime. That's so awkward, right? With this square, something to stick This four out here is I have room for my big dramatic our prime. So setting these equal we want to solve for E So, um e we're just gonna you can cancel out, ah, tooth of two factors of our prime square. So we just have our prime squared on this side, and we just want to bring over the four, so Oh, we cancel a pious Well, so then we have, um, Rhona over for a and then we have our prime squared, and then we can double check the units. So Rose, charge over volume volumes are cubed. Who's then? We'll have our got are huge times with our scored in the top is one over r and then the A s units of distance as well. So that will give you the are cube. You are square that we're looking for on the bottom. Actually, it is kind of weird that this doesn't have the f. Salah, not Oh, I don't know why I keep forgetting the absolute Not so. Yeah. We definitely needed Nestle on Not down here. Make sure. Yeah. So yeah. Look said

Okay, This probably have a solid insulated here, and we know that the density of this is fear. Various with whole equals. A are square. So that's a question that defines the density off this fear. And we have to calculate the electric field inside and outside this fear. We know that this fear has radios are okay, so it's just are and we must calculate the electric field in both situations. OK, so let's begin first, uh, part A. Let's calculate the magnitude of the electric Fuge when the radius is outside. So the readers is greater than the Raiders or kiss off the insulated sphere. We must use the got slow. And we know that the go slow is just when the electric field is uniform is just electric fields that multiplies the surface area of the Goshen surface. That is equal the charge inside, divided by absolute here. So it is the charge inside the guy ocean surface. And in this first configuration, we want that the radius of the ocean surface be greater than the Raiders of this fear. Therefore, we can see that charge inside is going to be the entire charge. Let's circulate this And so So we have here. Uh, the electric feud is going to be equal charge Q divided by the surface area, which is four pi are square, absolute zero. But since we must prove that the result is some equation that the program gives us, let's used that the charge que is going to be just density the multiplies, the volume of this fear Because the volume of the darshan here is the volume off the fear invited by four by absolute zero are square. We know that the density is described by desecration and the volume of this fear is also know Therefore we have that or barked A and start is just the electric field is going to be equal A which is the constant the multiplies our square Just the radios off the entire charge that city charge Ah, multiply by the volume and the volume is just for by divided by tree our cubic So we have here are four by absolute zero are square You can cross the four by and finally our answer to the electric field is going to be a r to the power or five you violated by tree Oh, are square, which is precisely what we must show Now let's calculate the bar should be fourth B. We must calculate the electric field when their radios of big ocean surface is smaller than the radius off the off the insulated sphere. Well, to calculate this were must used the definition right now. So let's put here again. We have the electric field there multiplies the surface area big ocean surface for by our square is going to be equal to charge inside the guy ocean surface divided by absolute zero. The problem this time is that it is. Since we are calculated insides, we're not taking the entire charge before. We must make your, uh, integral or hope Devi divided by absolute zero. We know what it's ho ho is just a Our square was the integral off a are square devi And if volume, as we know, is just the surface area which is four pi r square multiply by the radius reaches the are you've argued by absolute zero so we can simplify this. We can cross the fourth by where this work by in here And our equation now is just eat our square that is equal. Let me see a divided by absolute zero off the integral All our let's see off our to the power or 54 sorry r to the power of four me are Therefore, if we make this integral, we're going to find any divided by absolute zero are five divided by five The limits of integration goes from zero to our or forgot to mention that because we are not taking the entire the n virus fear. So finally, the answer for the electric foods is just okay. Divided by five absolute zero R to the power off three. Okay. And that's the final answer. Thanks for watching.