00:01
Does a point charge create a magnetic field? well, you can calculate one, but realize that it won't be a persistent magnetic field unless you have a stream of charges or current.
00:17
But we could go ahead and use the bios of art law to calculate the field of a point charge moving.
00:26
And it looks very similar to the electric field, the kulam's law.
00:32
So you can think of the biots of art law as working from the source of a magnetic field, which is a charge times of velocity.
00:43
And the direction of that field is a little bit more complicated.
00:48
It is a cross -product between the velocity and the r -hat, which we'll talk about what that r hat is.
01:01
So our first situation is we need to calculate the magnetic field due to two moving charges, a proton and electron, at the origin.
01:13
So that is our observation point.
01:20
And we want to find the total magnetic field which is coming from two separate charges, the positive and the negative moving charges magnetic fields just add or superpose.
01:35
Now, be aware that what we mean by that r -hat is it is a unit vector that points from the source to the observation point.
01:56
So i can show what that looks like if we are looking at the origin.
02:05
There are two different r -hats.
02:08
The electron r -hat points to the my -hat.
02:12
If we adopt the usual convention, the proton points to the negative x hat.
02:27
So if the observation point were directly in line with the velocity, there would be no magnetic field in that direction.
02:37
Yeah, we'll just choose the usual conventions, x the right, y, upwards.
02:45
So let's figure out these magnetic fields separately.
02:55
We'll find the magnitude.
02:58
The mu -naut, by the way, is a constant equal to 4 pi times 10 to the minus 7th in the si system, and will cause the magnetic field to be in teslas.
03:16
So let's find the magnitude times the velocity, 735, times 10 to the third, meters per second.
03:36
And then we have the separation between the particle and the origin, which is for nanometers, nano is 10 to the minus 9.
03:53
And the direction, we have a positive charge.
03:57
The v is in the minus y hat and the r is in the minus x hat.
04:04
And the r is in the minus x hat.
04:12
And we can simplify things.
04:15
Y hat cross x hat is negative z hat.
04:24
So our final magnetic field is 7 .35 times 10 to the minus for tesla in the minus z hat direction.
05:01
Check the exponent.
05:02
Yep, the exponent sounds good.
05:05
All righty, let's do the b minus.
05:09
Now that we've done that part, it should not be.
05:12
Too hard.
05:14
It's pretty much the same numbers, except the separation is 5 nanometers, and the vector part is a little different.
05:31
Okay, here the v is in the minus x direction, and the r -hat direction is in the minus y.
05:44
And we have a negative charge.
05:47
Lots of negatives.
05:55
And so we want to cancel x -cross -y is z -hat.
05:59
But we have a negative charge.
06:00
The negative due to the electron charge.
06:07
And so we still get in the minus z hat direction.
06:16
And let's see, we get 4 .7 times 10 to the minus 4 tesla.
06:25
And if we add up those in a vector sense, they are both in the same direction.
06:30
So a total is fairly easy to work out.
06:46
Okay, so that was kind of tedious.
06:47
The next thing we're doing is even a little bit harder, and that is we are going to have to figure out the magnetic and electric forces due to the electron at the proton, and a reminder of how that works out.
07:06
Okay, find total magnetic force on proton, find electric force on the proton, from the electron.
07:29
Reminder that magnetic force is equal to qv cross b.
07:38
And so we'll have to find the magnetic field at the proton due to the electron.
07:44
The electric force looks like k, which is one over four pi epsilon knot, q squared over r squared r hat.
08:00
And that's an attraction between the two along the line joining them.
08:08
But we're first going to have to find the magnetic field due to the electron at the proton.
08:19
And unfortunately, that is something different than we've found, but hopefully we've gotten a little bit of experience with the biots of art law...