A 0.360 -m-long metal bar is pulled to the left by an...

A 0.360 -m-long metal bar is pulled to the left by an applied force $F$. The bar rides on parallel metal rails connected through a $45.0 \Omega$ resistor, as shown in Fig. E29.29, so the apparatus makes a complete circuit. You can ignore the resistance of the bar and rails. The circuit is in a uniform $0.650 \mathrm{~T}$ magnetic field that is directed out of the plane of the figure. At the instant when the bar is moving to the left at $5.90 \mathrm{~m} / \mathrm{s},$ (a) is the induced current in the circuit clockwise or counterclockwise and (b) what is the rate at which the applied force is doing work on the bar?

Key Concepts

Motional Electromotive Force

This concept explains how a conductor moving in a magnetic field experiences an induced potential difference due to the magnetic force acting on its charge carriers. The magnitude of this induced emf depends on the magnetic field strength, the length of the conductor, and its velocity. It is a manifestation of Faraday’s law in situations where the motion of the conductor itself creates a changing magnetic flux.

Lenz’s Law

Lenz’s Law provides the direction of the induced current resulting from electromagnetic induction. It states that the induced current flows in such a way as to oppose the change in magnetic flux that produced it. This opposition is a natural consequence ensuring the conservation of energy in the system.

Lorentz Force

The Lorentz force is the force experienced by charged particles moving in a magnetic field, given by the cross product of the velocity and the magnetic field. For a current-carrying conductor, this force results in a net force on the conductor. This is crucial in understanding how an applied force must overcome the magnetic force that opposes the conductor’s motion.

Mechanical Power and Energy Conversion

This concept relates the work done by forces acting on moving objects to the conversion of mechanical energy into electrical energy. Specifically, the rate at which work is done on a moving conductor by an external force is equal to the power input, often calculated as the product of the force and the velocity. In electromagnetic induction problems, this power input is typically converted into electrical energy, which may then be dissipated as heat in circuit resistances.

Transcript

00:01 Okay, so we're doing chapter 29, problem 31.

00:06 So in this problem, we have a circuit with a resistor here, and these two ends of the circuit, these two wires are connected by a metal bar.

00:18 Let's draw that a little bit better.

00:21 So they're connected by a metal bar here.

00:25 And this is all in a b field that is pointing out of the page, sorry.

00:32 So b field out of the page.

00:37 This is the length l and they tell us it is moving towards this bar is moving towards the left.

00:44 Okay, so let's write some of these things down.

00:47 This is 0 .36 meters is the length of the bar.

00:51 It tells us the resistance of the resistor is 45 oms.

00:56 And it tells us the magnitude of the b field is 0 .65.

01:02 6 -5 tesla.

01:07 Okay, so we want to figure out the direction of the induced current in the circuit.

01:14 So to do that, we use linz's law.

01:17 And lindsay's law is the one that says the induced current produces a b -field that opposes the change in magnetic flux.

01:25 Okay, so let's look at this flux here for a second.

01:29 You know the magnetic flux is the b -field times the area.

01:33 So the b field is constant, right? so this is constant, but as the velocity moves to the left, this loop of our circuit is decreasing.

01:46 So that means our flux is decreasing.

01:49 So if our flux is out of the page and decreasing, to oppose that, the induced magnetic field must be out of the page.

02:00 Okay.

02:02 So now we've got to use the right hand rule and figure out which direction of a current...

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