0:00
All right.
00:01
So in this expression, we're given the radiative energy loss of a charged particle as a unit of time per unit time.
00:14
And so first we want to see if this makes sense.
00:19
So the units of q are coulums, units of acceleration, are meters per second squared.
00:32
And this is all going to be in si units.
00:35
The units of epsilon not are in coulomes for energy times meter.
00:47
And the units of c are meters per second.
00:55
So putting all this in, the units of det, d .t, r and 6 pi obviously doesn't have any units because it's just a number.
01:08
So this is going to be c squared multiplied by meters per second squared squared over kulom's squared.
01:27
Sorry, this should be squared.
01:29
Kulom squared for energy times meter and then meters per second.
01:43
So let's start getting rid of some stuff.
01:45
So immediately we can see that these cancel.
01:49
We can also see, so if we rewrite this now, this is meters per second squared times one over seconds.
02:03
This is now one for energy times meters, meters per second cubed.
02:18
We can see that this turns this into just a one.
02:24
And so if we flip everything that should be on the top, on the top, we get units of energy times meters times seconds over meters times seconds, over meters times second squared.
02:49
So this consists this, this case is this, and we get units of energy per second, which is what we would expect from a time derivative of energy.
03:05
So now we want to know if a proton with kinetic energy 6 m .e .v travels in a circular radius of 0 .75 meters, what fraction of its k .e.
03:59
Does it radiate per second? and so what we can do is we can use the fact that for circular orbits, a equals v squared over r.
04:25
To find our a, we can use the expression.
04:33
And this is just to make sure, as we don't know whether or not this proton is moving relativistically, it shouldn't be because that's a very, that's a relatively small energy.
04:44
Compared to the breast mass energy.
04:46
But assuming that we don't know that, we can use p equals gamma mv to find the to find the velocity where gamma equals 1 over the square root of 1 minus v squared over c not b squared over b squared, v squared over c squared.
05:14
And this will give us velocity since we know that p so from e squared equals m squared c to the fourth plus p squared c squared.
05:33
This gives us that our kinetic energy squared is equal to p squared c squared, which gives us that our momentum is going to be 6 mab per c.
05:52
And we'll see why those units are useful in a second.
05:56
So 6 m .ev for c is going to equal gamma, so 1 over square root of 1 minus v squared over c squared, multiplied by the mass of a proton, which is 938 .5 mep for c squared.
06:24
So you can see why we're using these units now.
06:26
It makes everything much easier.
06:30
By the velocity of our proton.
06:35
So if we divide six, divided by our rest mass, gives us 0 .0064, and that's multiplied by c equals b over the square root of 1 minus b squared over c squared.
07:02
So squaring both sides, so if we square this side, we square this side, we're going to get 4 .09 times 10 to the negative fifth, c squared, equals b squared over 1 minus v squared over c squared.
07:34
So multiplying this over, you get 4 .09 times 10 to the negative fifth, c squared, 1 minus v squared over c squared equals b squared so that's going to equal 4 .09 just expanding out our expanding this term here has 4 times using negative 5th b squared equals v squared so then combining like terms we're going to have 4 .09 e to the negative 5th c squared is going to equal 1 plus 4 .09, 10 to the negative 5th v squared.
08:26
And so if we divide both sides, we're going to get 4 .09, e to the negative 5th, c squared over 1 plus 4 .09, e to the negative 5th equals v squared.
08:51
And then taking the square root of both sides, we can get that v equals 0 .0064c, right? so non -relativistic.
09:12
And if we use the fact that c equals 3 times 10 to the 8th meters per second, and we can get a speed of the velocity of our proton equals 1 .92 times 10 times 10.
09:39
To the 6th meters per second.
09:43
So still relatively fast for our experience.
09:47
And then we can put into a equals v squared over r.
09:54
The radius, right, for our acceleration, it's going to equal 1 .92 times 10 to the 6th squared over our radius, which is 0 .75.
10:11
And that's going to equal 4 .90.
10:24
Times 10 to 12 meters per second squared...