The conducting rod a b shown in Figure 21.61 makes...

The conducting rod $a b$ shown in Figure 21.61 makes frictionless contact with metal rails $c a$ and $d b .$ The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. (a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed 7.50 $\mathrm{m} / \mathrm{s}$ . (b) In what direction does the current flow in the rod? (c) If the resistance of the circuit $a b d c$ is a constant 1.50$\Omega$ , find the magnitude and direction of the force required to keep the rod moving to the right with a constant speed of 7.50 $\mathrm{m} / \mathrm{s}$ .

Key Concepts

Faraday’s Law of Induction

This law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit. It is fundamental in understanding how moving conductors in magnetic fields or varying magnetic fields can generate electrical energy, forming the basis for many electrical generators and transformers.

Motional EMF

Motional emf refers to the voltage generated when a conductor moves through a magnetic field. The emf is proportional to the magnetic field strength, the length of the conductor within the field, and the velocity at which the conductor moves. This concept explains the creation of a potential difference along the moving rod in the presence of a uniform magnetic field.

Lenz’s Law

Lenz’s Law determines the direction of the induced current resulting from electromagnetic induction. It states that the induced current will flow in such a direction that its own magnetic field opposes the change in the original magnetic flux. This principle is essential in predicting the behavior of induced currents in circuits under changing magnetic conditions.

Lorentz Force

The Lorentz force is the force experienced by a current-carrying conductor in a magnetic field. It is calculated as the cross product of the current element and the magnetic field, and it provides the mechanism by which mechanical forces act on electrical circuits. This is key for understanding how an external force is required to balance the magnetic drag on the moving conductor.

Ohm’s Law

Ohm’s Law relates the current flowing through a circuit to the voltage across it and the resistance of the circuit. In the context of electromagnetic induction, it is used to calculate the magnitude of the induced current from the induced emf and known resistance, allowing for the determination of forces and circuit behavior.

Transcript

00:01 So this problem, we have a conducting rod sliding on rails and a uniform magnetic field.

00:08 Our goal is going to be to find various things about this setup.

00:12 And on this depiction, we have our magnetic field going into the page.

00:16 Its value is 0 .8 tesla.

00:18 The rod is moving to the right at 7 .5 meters per second.

00:22 The height of this from c to d is 50 centimeters, and i'm denoting the length from d to b as l.

00:32 So the first question is to find the induced emf.

00:38 So we know that the induced emf is going to be given by our change in flux over our change in time.

00:47 We can also write our flux as b times a.

00:54 We don't have to deal with any cosine because the perpendicular of this square region of area and the magnetic field are in the same direction.

01:02 So that cosine is 1.

01:05 Our b field is constant, so we can pull that out.

01:09 So we have b, change in area over change in time.

01:15 Our area, we see that that is our h times our l.

01:19 So we can put in b, change in h, l.

01:26 And here, our height is not changing.

01:30 It's only the length from d to b that's changing.

01:32 So this is b, h, changing length over change in time.

01:39 And that change in length over change in time is just our velocity that we're given.

01:43 So we have all these numbers and we can see what this value is.

01:48 So our b is 0 .8 tesla.

01:53 Our height is 50 centimeters, let that into meters.

01:59 Our change in length over change in time is 7 .5 meters per second.

02:03 And if we plug all of this in, we get 3 volts.

02:14 Now that we have our induced dmf, the question is, what is the direction of the induced current? here we're going to need lenses law.

02:23 So lenses law tells us that the induced magnetic field establishes to mitigate the change in net magnetic flux.

02:49 So we see here, since this bar is moving over to the right, our magnetic flux from this external field into the page is growing...

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