Question 4 A sphere of radius R has charge density p(r) =...

Question 4 A sphere of radius R has charge density $p(r) = kr$. a. Compute the total charge of the sphere. (0.5 marks) b. Compute the electric field $\vec{E}$ at a distance $r$ from the center of the sphere. You should have one expression for $r < R$ and another for $r \ge R$. (1.5 marks) c. Using the expressions from part b), compute the total energy of the system. (3 marks)

Transcript

00:01 So we have a very a child who are people who are by getting our quality and we're going to be in a lot the area subject to as early street.

00:09 And we're going to find out of the area to keep in the charge by the school.

00:14 So we know that we can write as a as a charge due delany.

00:19 The part is the one charles day of going to write this as a re -boat meeting a and then gave it was a website.

00:26 So it says, every year is a history.

00:28 I hope that although not i'm probably going to have been to be able to put a lot of him in to and then would you get in the way of the great and very long.

00:41 Do you what you may think of this for this year, i'm going to be in there and the most exciting people and the community and be part of the guys i'm going to criticize to you this is a very important r power 3n and this is r power n plus 2.

01:04 Vr and this is a sign of teta vita and deviation of d4 5 is a little over 025 teta is a little from 05 and r &1 from 0000 3.

01:18 Right purpose is q and the pervincar radius and then it's very quick for you limit of teta c22 and the level of 5, as you know how it is the 0 .0 to 2 file.

01:33 But then the opportunity is actually can be applied by the other and this is infertive power every type of n plus kilder and plus kinti and plus k.

01:46 And it's in 5.

01:49 So we go in all so we're going to find out to the capital fala in him and then c1m so we think the value of energy that's the radius of area.

02:07 So the second part of this problem is that we need to find the value of n such that the value of electricity at a distance r by 2nda is equal to the electric field a at 4 hour distance.

02:24 So for this we need to find the value of n.

02:29 So let's call it.

02:32 We have given this total charge and that we have given this volume charge density.

02:39 So first we need to be a little bit at itself boat.

02:44 So it can keep inside format.

02:53 So this is a sphere and we will draw a small portion of surface inside and this is a that's a resistance called off so we were here to be d d d ds as we can still cue in ghost so you can write this t and grays as 4 dot bd b b divided by xanot so this is three times four five are related to nal r squared and is equal to a robot so this is always basically moving outside divided by r2 power n of self degregation r2 power l plus 2 dgr degation sign with a dgr dgf related by svelinot so this is no lot and so you're not in 5 power 10 and this is known from and the total let's tell you first then you know how to change so here we're that in r2 % of 3 million plus 3.

04:05 And this is 8 times to be called by r square.

04:10 Right now, this pie and this pile tensile.

04:15 This 2 % is formed by 3 times.

04:19 So we need certain amount to do it in a bit in that we are not r5 % and this is r5 % and this is r5 plus 1.

04:30 Divided by 10 plus 3.

04:35 Right.

04:36 This is the value of electrical left inside point.

04:40 Now for the outside point for outside point, this whole sphere will work as the point charged...

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