The drawing shows two long, straight wires that are suspended from a ceiling. The mass per unit

The drawing shows two long, straight wires that are...

Key Concepts

Magnetic Force Between Parallel Currents

This concept describes the interaction between current-carrying wires. When two wires carry currents, they create magnetic fields around them. Depending on the direction of the currents, these wires can attract or repel each other. The force per unit length between two long, straight wires is given by a formula derived from Ampère’s law, which is fundamental to understanding electromagnetic interactions in current-carrying conductors.

Magnetic Field of a Long Straight Wire

A long straight wire carrying an electric current produces a magnetic field that circulates around the wire. The magnitude of this field at a given distance from the wire can be calculated using the Biot-Savart law or Ampère’s law. This field is crucial in determining the magnetic force experienced by another nearby current-carrying wire.

Force Equilibrium in Mechanical Systems

In many physics problems, including those involving suspended wires, it is important to analyze the equilibrium of forces. The wires are subject to gravitational forces and tension from the strings that suspend them. When the wires experience an additional force (such as magnetic repulsion), the system reaches a new equilibrium, and the forces are balanced. The equilibrium condition is used to relate the magnetic force to the components of the tension in the supporting strings.

Trigonometric Resolution of Forces

This concept involves breaking down forces into their horizontal and vertical components using trigonometry. In the context of suspended wires, the angles of the supporting strings allow the determination of how much of the tension is contributing to counteracting the magnetic force versus balancing the weight of the wire. This resolution is key to setting up the equations that lead to finding the current in each wire.

Transcript

0:00 Hello.

00:01 So in this problem we have two current carrying wires carrying currents in the opposite direction and they're being suspended by ropes.

00:11 Now because they're carrying currents in the opposite direction, right, each of those wires creates their own magnetic field and then the magnetic field from one wire interacts with the current in the other wire and these result in magnetic forces.

00:24 And we know that if wires carry currents in opposite directions, they're going to cause repulsions.

00:30 Forces.

00:31 So overall we have a situation that looks something like this, right? so we have wires being suspended from the ceiling by ropes.

00:44 And because the distance between the wires is going to be important, i'm going to draw this line right here, and then i'm going to draw our wires being suspended from these ropes.

00:54 Let's call the wire on the left wire one, and we'll have that current going into the page here, represented by an x and the current will have a magnitude i this wire here will be wire two it'll be carrying the current out of the screen we know from the problem that the current is the same value so the same magnitude of current each but of course they're in the opposite direction so we have the dot here representing the current going out here okay we also know the links of these wires here and we also know the entire angle here and we'll call that angle we'll call that angle theta, right? and we know that it's going to be 15 degrees.

01:35 We also know this length little out here is 1 .2 meters.

01:39 Now it's going to be nice and useful to bisect this angle, right, so that we know that these make two right triangles, each with half of theta as an angle.

01:50 There we go.

01:52 Two right triangles here.

01:54 Okay, another important factor or a variable we're going to need for this problem is the distance between the two wires and we know that that's represented as this line i've drawn here, the bottom of this triangle, and so it's going to be a distance of r.

02:13 Let me instead, let me better represent that.

02:16 So this entire distance between the two wires is r here.

02:20 Okay, good to know.

02:23 And we can also see, since we're going to know this length here, we can tell what that's going to be by by looking at the equation, or just doing some trigonometry here for these two triangles i've just drawn.

02:37 So if i wanna know how long this length of this triangle is here, since this is a right triangle and the angle opposite to it is one half of theta, and i know the length of half of this r here is going to be l times sine of the theta we're given over two.

02:56 So really it's sign of 7 .5 degrees, which is half of 50, anyway, we've done a little bit of geometry to set up this problem.

03:05 Now really what we're going to need to know in order to figure out the currents these wires is the forces acting on these wires.

03:11 Because what's happening, right, is that not only is gravity trying to weigh these wires down because they are massive wires, the magnetic force between the two wires are repelling each other.

03:23 So now what we're going to do is think about what are the forces acting on our wires and then in what directions.

03:31 So why don't we just do a little bit of thinking here? so let's just focus on wire 2.

03:37 Because they carry the same current, we can figure out the current in each just by solving for the current in 1.

03:43 So let's go ahead and work on that.

03:46 So for our wire 2, we know that's going to be experiencing a gravitational force straight down.

03:53 We'll call it f subg, and that's going to be of magnitude m times g of the mass of this wire multiplied by the acceleration due to gravity.

04:07 Now, we also know that the wire is going to feel a repulsive magnetic force from the other wire directly to the right here.

04:17 And we'll call that the magnetic force of wire 1 on wire 2.

04:25 Now, what is that going to be given by? right, well, it's going to be given by the current going through wire 1 times the length of wire 1 multiplied by the magnetic field created by, sorry, let me say that again.

04:40 This is the force of wire 1 acting on wire 2.

04:44 So it's going to be given by the current going through wire 1, the length of wire 1, and this.

04:51 Sorry, let me just say that again.

04:53 This is the force of wire 1 on wire 2.

04:56 So this is actually going to be the current flowing through wire 2, multiplied by the length of wire 2, multiply by the current through the magnetic field created by wire 1.

05:07 Now, we also need to multiply this by sign of the angle between the magnetic field created by wire 1 and the current direction in wire 2.

05:20 The magnetic field created by wire 1, given our right hand rule, if we put her thumb into the screen along the direction.

05:27 Of the current, then our magnetic fields, like our fingers, are going to curl around in a clockwise.

05:33 Oops, let me rewrite that, in a clockwise manner.

05:37 That means at the location of wire 2, the magnetic field, due to wire 1, is actually going to be headed straight down.

05:48 And it's going to have a magnitude of mu not times the current through wire 1, multiplied by two, i mean divided by two pi times r, which is the distance between wire 1 and wire 2.

06:01 Okay, good to know.

06:04 Now, if magnetic field, the magnetic field created by wire 1 is headed straight down, and the current in wire 2 is headed straight out, then those two things are perpendicular.

06:15 If those two vectors are perpendicular, phi is then 90 degrees and sign of phi those to 1.

06:21 So that we know that the magnitude of the magnetic force acting on wire two caused by the magnetic field created by the current in wire one is given by in magnitude by the current through wire two the length of wire two and the magnetic field caused by wire one and we know what that is so we can plug that we know that that is it's right here so we can plug that in so we have that the force acting on wire two from wire one is given as i l times mu not times i all over two pi times how far apart the wires are from each other.

07:01 Now when i simplify this out, i'm just going to group a few terms here.

07:05 I'm going to see that the magnetic force acting on wire two from wire one by magnitude is mu not divided by two pi multiplied by the length of wire two, which they're both the same length, so the length of our wires, divided by how far apart our wires are, multiplied by the currents squared.

07:24 So the current in each multiplied together, but they have the same currents, so it's the current squared.

07:28 So we have the magnetic force acting on wire two, the magnetic force acting on wire two, and what else is acting on wire two? well, of course, remember, we have a tension heading up into the left.

07:43 So that's going to be, we'll label it as f sub t for the tension, and that's going to be heading up in that direction.

07:52 Like so.

07:59 Okay, so.

08:00 Now what we want to do, now that we figured out what the forces are acting on our wire, we want to rewrite this as a free body diagram.

08:10 So what i'm going to do is use this room at the bottom to draw a free body diagram for our wire two.

08:17 And then we can figure out newton's dynamics based on what's happening in this problem, and then from there, through newton's laws, we can figure out the current through the wires.

08:29 Now, let's draw a free by diagram.

08:31 So let's represent our wire, wire 2, as just a dot.

08:36 We know that the force of gravity heads straight down of magnitude m times g...

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