00:01
Greetings and salutations.
00:02
In this question, we're given this setup.
00:04
We have two current carrying wires carrying currents into the page and they form the bottom two vertices of an equilateral triangle with a side length of 2 .5 centimeters.
00:14
We're told the current's the same and it's 12 amps for both and we're asked in part a, what is the magnetic field at the third quarter of the triangle? so let's go ahead and figure out our directions here first before we do anything else.
00:25
Take my right hand, point my thumb in the direction of the current.
00:28
So for the blue wire, it's going to be into the page.
00:31
I'm going to curl my fingers and the direction that my fingers are pointing when i get up to the top is the direction that my magnetic field is going to be pointing.
00:40
So that's going to be up and right and then the same thing will happen if i put my thumb in for the red wire.
00:46
Thumb in, curl my fingers, it's going to be down in.
00:49
So this is going to be b, i'll call it b1 and i'll call this other one b2, which just corresponds to the colors of the wires there.
00:58
Now this is going to be a very important point.
01:01
We need to know which direction these are pointing in.
01:05
Well because this is an equilateral triangle, that'll be pretty easy to figure out.
01:10
So our magnetic field in the blue is going to be completely normal to this vertice here, this line, and our magnetic field in the red is going to be completely normal to this vertices, so this line.
01:23
I haven't drawn them quite straight but we're going to have two vectors at a right angle to one another there.
01:31
B1 points along the one vertice, b2 points orthogonal to that.
01:36
That means that i know that in the positive x direction i'm going to split this into two 45 degree angles here.
01:45
Because it's 90, i'm splitting it perfectly in half because of symmetry, so i'm going to have two 45 degree angles.
01:50
So our net magnetic field as a vector is going to be b1 plus b2 as vectors.
01:57
In this case, that means we're going to have x and y components.
02:00
However, our magnetic field in the y direction is going to be equal to zero.
02:06
Hopefully you can see that because of symmetry.
02:08
Because these have the same current and they're the same distance l away, and one of them is pointing 45 degrees up, the other is pointing 45 degrees down, those y components are going to cancel.
02:19
So all we're going to be left with is a magnetic field in the x direction, and that's going to be two times the field from either of these in the x direction.
02:28
So i'll call it b1x here.
02:30
And that's again because of symmetry.
02:31
So what is that magnetic field going to be? well, it's coming from a wire, so it's going to have four mu naught times i over two pi times how far away we are, which in this case is l.
02:41
And so that's all it's going to be...