00:01
All right, so let's say we have a sphere that is insulating and has a total charge q spread throughout the volume.
00:09
Then we want to find the electric field inside the sphere.
00:12
So if you use galsalsala, we'll have e times 4 pi r squared is equal to the charge enclosed, which we can write as q over the total volume.
00:23
And then we'll have a one of our epsilon not here times like the volume of our volume of our little.
00:32
Gaussian surface.
00:33
So the four -thirds pies are going to cancel.
00:36
And what we'll ultimately arrive at is that this is just k times q times r over r cubed.
00:44
So that's for r less than r.
00:47
And then outside the sphere, our total and close charge is just q.
00:50
So we'll have e times four pi r squared equals q over epsilon not.
00:56
So of course e is going to be just k kk over r squared.
01:03
And then part c, we want to find the voltage inside the sphere.
01:08
So the voltage going from a point very far away, we'll just say infinity to some point r of e, you know, dot, dr, we'll call it r prime.
01:23
This is going to be the integral from, right, it is infinity to r to the surface of e, d r minus the integral from r to little r of you know this and so we'll have different electric field values here so for a point charge this will be you know negative this will be we can switch the limits really quickly so we'll write this from r to infinity and this is going to be k times q times r over r squared.
01:59
And then for the second term, we will have minus k times q over rq times the integral from r to r of r, dr, r prime, d r prime, i guess we'll call it.
02:14
So this first integral is going to just give us negative kq over capital r.
02:23
Or actually, it'll be a positive.
02:25
And then for our second one we'll have kk over r cubed times one half are we'll write it this way yet one half r squared minus one half capital r squared like this and so we can rewrite this as k times q times one over r minus r squared over two r cubed minus one half actually plus one half over r i think like this and so if we rewrite this this should be k times q times three halves r minus r squared over two r cubed so we can write this as k times q over two r let me just make sure i'm not forgetting a number anywhere.
03:31
I think we'll have an r cubed, and then we'll have 3r squared minus r squared.
03:38
So that's our voltage inside.
03:39
Outside, it's very simple because we've kind of already done this...