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2) A sphere Of radius Rhas total chargeThe volume charge density within the sphere isP = Pwhich varies linearly from Pin the center of the sphere to zero at the edg...

Question

2) A sphere Of radius Rhas total chargeThe volume charge density within the sphere isP = Pwhich varies linearly from Pin the center of the sphere to zero at the edge: Show that P =30 TR' Find the electric field at some radius inside the sphere: Check that has the correct value when r = R

2) A sphere Of radius Rhas total charge The volume charge density within the sphere is P = P which varies linearly from Pin the center of the sphere to zero at the edge: Show that P =30 TR' Find the electric field at some radius inside the sphere: Check that has the correct value when r = R



Answers

A sphere of radius $R$ has total charge $Q .$ The volume charge density $\left(\mathrm{C} / \mathrm{m}^{3}\right)$ within the sphere is $$\rho=\rho_{0}\left(1-\frac{r}{R}\right)$$
This charge density decreases linearly from $\rho_{0}$ at the center to zero at the edge of the sphere. a. Show that $\rho_{0}=3 Q / \pi R^{3}$ Hint: You'll need to do a volume integral. b. Show that the electric field inside the sphere points radially outward with magnitude $$E=\frac{Q r}{4 \pi \epsilon_{0} R^{3}}\left(4-3 \frac{r}{R}\right).$$
c. Show that your result of part b has the expected value at $r=R.$

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 what here we have a non conducting a spare. Okay. So you want to find the entire sergeant closed in? So the first question we know that the entire size clothes in the surface will be nothing but to equal to uh you got anything but the density At all? In two DB. Okay. D. V. And from zero to what did you go to our? Okay. And over here we have the density with the really dependent on the radial distances role as upon. Arab on our. Okay, so equal to uh oh is upon our upon. Okay. Are the release of the sphere? And r is the distance from the center? Okay. So basically what we can do over here is that we can just to get by putting violence is equal to zero to our throw us upon. Arab on our in two D. V. So did he believe that if we go from this is a sphere? So we if you go from our distance from center. Okay. And we cut the element of the arctic nous. Okay, so what really? The volume of this one? This area into the heart of the village, That means to buy our into the r right to buy our into the R. Or rather we can say the volume, this is the area. So this will be, the volume will be, this is a spherical spherical wavefront. Have the volume uh area is 40 R squared and the techniques tr so we can write you a DVD, we will look for pi R squared into D. R. Okay, so this issue comes out to be nothing but this is constant. This is also concerned the role as upon hard and four Pi Johnson comes out over here. So this will be due to add, this is our this is our our queue, they are. So since we can calculate from here that this will be wrote us upon our into four pi into our to develop for about four one that anyone are you can send out. So over here this comes out to be raw row. This forum. This forest also gets a role as by our cuba without recharge inside. Okay, what is the next part? So in the next part we need to find that we need to find that the and that gives them into the medical field inside the spare. So basically, uh, we know that electric field inside of a magnitude delivered. But the girls now we know that electric field E and if you feel even do nothing but and to feel even way uh, electric flux. Even when I think what you inside upon absolute Lord. Okay, so like the blocks have into area equal to Q. Inside upon absolute north. Okay, so you can see that uh First of all we will calculate the Q. Enclosed in the any art area. Okay so how are you going to find that? So we can find that with the same logic that you include. He called to 0 to our. Okay. 0 to not the entire and zero to just any. And Abrar one row. No Aragon are into again for by our Squire. They are. So this will be run over a four paragon or upon our And our cue. So r. to the power of four upon 4 I guess. Okay so the Senate foreign forget can send out this comes out to be Pie runner into A. To the power of four upon our I don't give out of four upon us. Yeah. Okay. Okay. So now from here in a complete E we can write it into four pi R squared equal to Q employees that is pie Are known to R to the power of four upon our Okay, I got the power of four upon our and this agent includes about excellent or rights upon absolute north. So these are and one to our militancy lord and the pie and pilot considered as well. So the electricity comes out to the runner our Squire upon four times of our lives. Don't know. Okay, four times of Arabs in North and over the head and over here we can multiply it by Yeah, over here. What we are missing over at that age. So when we do multiply this, this comes out towards him and

Ah, we choose a Gaussian surface of radius r, which is less than are not in concentrate Witty spear we calculated of electrically losing goes a lot black. The judge density the row e and it is given us row is a call to take you up on TV. Given that the charge and it is a function of the radial position, there is a role he R is equal to Rho or not are upon are not so the charge enclosed is equal to limit low e dizzy So we have zero to r ro, not hi dash upon are not or by our dish square the our dish. So Cuban closes or or not I r to the power for upon her or not Take this as equation one. This equation refers to the charge and closed So now, to find a total child assume eyes equal to earn no are not and the charge and city can be calculated using the formula Q role by are not to the power for upon are not so. That is a role not our but it will not buy are not cube so raw not will be queue up on by are not cube this situation two. So now substitute substitute the value of right or not from equation too. Immigration one we get you enclosed R is equal to a row, not by artists Power four upon raw notes Oh, that is a girl took you are upon are not to the power for, um Does the electric field everywhere with thin does feel is found using well this loss Now the the air is equal to Q and closed upon Absalon Not so that is the for by hi square is equal to Q Upon absolute note, Q Q are not to be power for but this implies he is equal to kill upon for by epsilon not I square upon are not to the power for Does the equation for electric field within the Yeah, in terms of kill are not an I And since the Josh distribution is positive, the electric field blind radial e outwards

Okay, so we're in Chapter 23. Problem 21 here. So says the volume charged density wrote e within a sphere of radius are not It's just me. Then I wrote the function of our is not are one minus R over r not squared. So this is our function for row E and where are is measured from the center of this year, ro not as a cost four point p inside the spear inside this year. So that means are less than are not determine the electric potential the and the infinity equals zero. Okay, so we first need to find the electric field. Since the charge distribution is fiercely since symmetric, we know the unused Alison's law where e dot d A is just the times of surface area of a sphere. Miss equals the charge and closed over. Absolutely not. So the charge and clothes now given as 4/3 pi r cubed times wrote e of are all over absolutely not. The total value for third pie are not cute. Okay, but this is actually not the case. Actually, we can't just take the ratio of the volumes, your times the volume density, because It's not constant. It was constantly could do that. But now it depends on our. So, actually, we have to take another step here. So we need to calculate what charge and closed is and this is given. But when we don't have a constant row E oven integral. We want to integrate from zero toe are row e of a function of our d. Okay, so let's do that. So charging clothes is integral from zero to Are we change? D v d r we're left with for pie are prime squared. And now we have ro not one minus our prime squared over r not square. Yeah. Okay, so let's solve this. We have or pie Rhona. Now we have r squared on the first becomes R cubed over three, and then we subtract off are to the fourth. So this is our fifth part of the five. Over five are not squared. So this is what our charging closed is. So let's take this back into our houses. All equation. And we should see that e is now one over for pie. Absolutely not One over r squared times for pi ro, not R cubed over three minus our five part of the fest over five are square. So it's cancelled out these four pies and some of these are there gonna cancer out. And we're left with the answer of of our e field being row not over. Absolutely are. Then we have our over three minus r cubed over five are not squared. Cool. So that's our e. Another thing we need to find out is what the total charges, which is the same as integrating what we had before from zero all the way to are. Not if we do that. We see something very similar to this equation, but it all simplifies out being ate pie. No, no. Are not Cube all over. Okay, we're almost there. We're almost have what we need. We know E for inside the sphere. We know what pew enclosed inside the sphere is a function of our is and we know the total. So now we're trying to figure out the potential O. R You can subtract off potential are not and see that this is the same as negative, Integral of ee dot The l from are not Oh, okay. So why don't we do this in a girl. So this becomes negative and a roll from Are not our of Rhona Over. Absolutely not. Are over three finest r cubed over five are not squared. We are. And let's calculate that, then this becomes negative. Ro, not over. Absolutely not. We have our squared over six minus R to the fourth over 20 are not swear Mrs Evaluated from are not our Okay, So what's out of page here? We're trying to solve he r which is be as a function of our less than on our This isn't how being ahed Plus the answer. We have time there. We haven't evaluated it yet, but we just had be not over. So this is B r equals R and r plus our answer from before. So we are. Let's figure out what Veena is. Well, Veena is just the potential from a point charge with the total charge. So this is que total over more pie Absolutely are not. So this is enough. So if we plug in what we got for Q total, we can simplify things more and this comes out to being to row, not are not squared over 15. Okay, Lastly, we plug our stuff, There's a view not And we add on our answer. All evaluated from or not are so now we can just plug this in must pull out our run on over Absolute merit We should see real efforts are not squared over four minus r squared over six plus part of the fourth over 20 are not cute. Oh, Miss equals the of our less than art out. Finally we got there lots of any girls, but we made it.


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