The two conducting rails in the drawing are tilted upward so they each make an angle of 30.0^∘ w

The two conducting rails in the drawing are tilted upward so...

Key Concepts

Motional Electromotive Force (emf)

This concept explains how a voltage (or emf) is induced when a conductor moves through a magnetic field. The free charges in the conductor experience a magnetic force that separates them, creating a potential difference along the length of the conductor. It forms the basis for understanding how mechanical motion can be converted into electrical energy.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force describes the force exerted on charges moving in a magnetic field. For a current-carrying conductor, this force is given by the cross product of the current element and the magnetic field, resulting in a force perpendicular to both. This principle is crucial for analyzing how magnetic fields can exert forces on conducting materials, thereby influencing their motion.

Force Equilibrium

In many physical systems, especially those in steady state or constant velocity, various forces balance each other. This concept involves setting the net force to zero in order to determine unknown quantities. In problems where motion is at constant velocity, it implies that the driving forces are exactly counteracted by the opposing forces, allowing for the application of Newton's laws.

Component Analysis on Inclined Planes

This method involves breaking down forces into components parallel and perpendicular to an inclined plane. For any object on an incline, gravity can be resolved into a component that drives motion along the plane and another component counteracted by normal forces. This analysis is essential for understanding the dynamics of objects moving on slopes, particularly when additional forces, such as magnetic forces, are present.

Transcript

0:00 Hello.

00:01 In this problem, which seems rather difficult, it's mostly going to be relying on the information we learned about the nature of magnetic forces for current carrying wires, and thinking back about newtonian physics and doing some newton second law analysis.

00:19 So we have this diagram in the book, but i'm going to draw it in different ways here for us to get a good idea of what is going on.

00:26 Right, so what's going to happen is this rod is going to start to slide down these two, these conducting rails.

00:33 And it's going to then start, because of the magnetic force, a current is going to start to flow.

00:37 So ultimately, we're going to be having, from a top view, so looking down on this thing, right? we're going to have this rod, and it's going to be sliding down the incline.

00:48 We'll say down the incline is to the left at some constant velocity.

00:53 Because as it starts to slide, some current begins to build up.

00:59 And then because our magnetic field in this case, if we're looking down on this, is heading out of the page, if we do that, then we see our right hand rule tells us that if our pointer finger follows the current and we rotate our wrist so that our middle finger is heading out of the page as the magnetic field is, then of course our thumb is heading back into the right.

01:23 So we have a magnetic force that is heading directly to the right in this case.

01:30 And that's going to be something that helps to counter the part of gravity that is pulling this thing down.

01:38 So we have some part of the gravitational force that is making this rod slide down the incline.

01:44 And then another part of the magnetic force that is trying to keep this rod from sliding down the incline.

01:56 So what we're going to do now is draw a free body diagram for this rod sliding down this incline and then do an etonian analysis on it.

02:07 So let's kind of redraw what's going on in the book but from a side view, right? so we have basically, though it's a rod, we're going to consider it an object sliding down an incline and we know the angle here is 30 degrees.

02:23 So we have an object here, sliding down an incline.

02:26 And so let's think about what the forces are going to be and in what directions.

02:30 Now, of course, we're going to have a normal force that is perpendicular to the surface like this, and our gravitational force is going to act directly downward.

02:38 Now, i'm not drawing the vectors in any particular links.

02:41 I'm mostly drawing them so that the diagram looks good.

02:44 Now, remember, so if we're sliding down, if we're sliding down, the rod is sliding down this way at this constant velocity, v, then that means the current is going to be heading into the page here.

02:57 So the rod's current is going to be heading into the page.

03:02 Now again, we have a magnetic field that is directly upward and a current that is directly inward.

03:10 When we do our right hand rule there, just like we saw that the magnetic field is directly to the right on our left diagram, that means, again, it's going to be directly right here.

03:19 So the magnetic force is trying to push this rod directly to the back of our incline.

03:28 Let me read that again, the back of our incline.

03:33 So that is our free body diagram.

03:36 Now what we need to do is do a little bit of balancing.

03:39 So we know because the velocity is constant along the incline, we know that the force, the net force of the net force along the incline must be zero.

03:59 The net force along this incline.

04:01 Must be zero.

04:02 So if we can figure out the components of the forces that are acting along the incline, we can balance those components and then hopefully solve for what we want to know, the current flowing through this wire, because we know the current is related to the magnetic force and the magnetic force is going to be part of the net force along the incline.

04:23 So let's try to break these vectors down into the components that are along the incline.

04:28 Let me just do a little redraw here to make things a little bit.

04:31 Nicer.

04:32 So if our incline is along this direction here, right, let's set up a coordinate system where one is along the incline, right? we want to balance all of our forces there...

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