00:01
In this problem, we're asked to find out the magnetic forces that are acting on each side of a triangular shape that is made out of wire that carries a particular current.
00:12
Now, it's going to look something like this where we have a triangle made out of wire.
00:19
We have a constant current going through that triangular wire.
00:25
We're also given an angle phi here.
00:28
And the length of this side here.
00:32
Now, we're going to label the corners, a, b, and c.
00:37
So therefore, this side on the bottom here is side a, b, and we can label it the length of side a, b, and that is given as two meters.
00:49
Now, we also know that there is a magnetic field directed directly to the right like this, and of course that is going to bring about the magnetic forces that are acting on the charges moving through this wire or because the current is flowing through the wire.
01:07
Now, in order to figure out the force on every particular side, we just have to follow our force equation.
01:14
The magnitude of our force will be given by the current flowing through the particular part of our triangle.
01:20
How long that side of the triangle is, the magnetic field magnitude, and sign of the angle between the magnetic field and the current direction.
01:30
Direction.
01:31
So we just need to apply this equation for every side of this triangle, and then just keep track of both the magnitudes of the forces on each side, on each wire, each side of the triangle, and think about the right -hand rule to figure out the direction of that force.
01:45
So let's first start with the bottom part of the triangle, or what we call side a, b.
01:54
Now, let's think about how this force equation here works for this particular side of our triangle.
02:00
So the magnitude of the magnetic force is going to be the current i going through that wire times the length of the wire, length a, b.
02:08
We know that value, i believe it is two meters.
02:13
And then we know the magnitude of the magnetic field through which the wire is, that the wire is suspended in, sorry.
02:22
And we need to figure out the angle that is between the current carrying direction and the magnetic field direction.
02:29
In the bottom part of this triangle, so i'll just highlight this side, right, the bottom side of the triangle, we see that the current is heading to the left and that our magnetic field is heading directly to the right.
02:41
So the angle between two completely opposite vectors, so directors that are in the opposite direction, the angle between those two is 180 degrees.
02:50
And sign of 180 degrees is actually zero.
02:53
And this makes sense because the current or the charge flow, the motion of the charges is directly, anti -parallel to the external magnetic field and therefore there's going to be no magnetic force acting on the bottom part of the wire so that's one part so it's only one part of our force and of course since the force is zero that means it really has no direction now let's think of the other sides let's think about the other sides of this triangle first let's think about the vertical side the vertical part or side a c and so that means we're going to be thinking about this side right here for now just to highlight it.
03:37
So let's write out our equation again.
03:39
Actually, real quick, before we write on our equation, we notice right that we're going to need the length, but we're only given the length ab, not the length ac.
03:48
So we need to use some trigonometry to figure that out.
03:51
If we look at this triangle, we have the angle phi, which is in the corner b.
03:58
So if we know this length ab as well, we can just use a tangent.
04:03
Function to figure out the side, the length ac.
04:06
So just doing a little trigonometry.
04:08
Length ac divided by length a, b, gives us the ratio, the trigonomic function of tangent.
04:16
And we know the angle here is given as 55 degrees.
04:20
Our theta here is 55 degrees.
04:25
So we now know how to express the length of side ac in terms of length of side a, b, which we know, and in terms of this angle, five.
04:36
So when we go and write out the magnitude of the force that's acting on this vertical side, we get the current times the length of side ac, times the magnetic field strength, times sign of the angle between the current direction and the magnetic field direction.
04:56
Now, it was easy up top for the bottom side.
04:58
The sign of 1a just made it zero.
04:59
We didn't have to plug in any new numbers, but now we're going to have to plug those in.
05:02
So our current value, the current flowing through the wire, is 4 .70 amps.
05:06
The length of the side is going to be the length of side a, b, or two meters, multiplied by tangent of 55 degrees, multiplied by the magnitude of the magnetic field, 1 .8 teslas, multiplied by sign of the angle between the current direction and the magnetic field direction.
05:25
If we go up to our drawing, we'll see that the current is heading up in this part of the wire, and the magnetic field direction is heading to the right.
05:31
So the angle between those two is actually just 90 degrees.
05:36
Plug a 90 degree in here, and we remember 90 degrees turns into, sign of 90 degrees is 1.
05:43
So we just plug all this into our calculator, and we end up getting that the magnitude of the force acting on the side ac of this triangle is 24 .2 newtons.
05:55
The question is, is what is the direction of this force? so we have to use the right hand rule.
06:00
So we put our pointer finger along the direction of the current like this.
06:04
We twist our wrist to make sure our middle finger.
06:06
Points in the direction of magnetic field to the left...