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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 $\mathrm{m} / \mathrm{s}$ .

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a. 3.00 $\mathrm{V}$b. According to Lenz's law, the current flows in the rod from $[b[$ to $] a]$c. 0.800 $\mathrm{N}$

Physics 102 Electricity and Magnetism

Chapter 21

Electromagnetic Induction

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Inductance

Alternating Current

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Hope College

University of Sheffield

Lectures

03:27

Electromagnetic induction is the production of an electromotive force (emf) across a conductor due to its dynamic interaction with a magnetic field. Michael Faraday is generally credited with the discovery of electromagnetic induction in 1831.

08:42

In physics, a magnetic field is a vector field that describes the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude (or strength); as such it is a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter (usually in the cgs system of units) and B is measured in teslas (SI units).

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The conducting rod $a b$ s…

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The conducting rod ab show…

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50.0 cmThe conducting …

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A conducting rod of mass $…

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The conducting rod shown i…

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A conducting rod with leng…

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Rod on Conducting Rails Tw…

this problem. We have a conducting rod sighting on rails in the uniform magnetic field. Argo is gonna be to find various things about this setup and on this depiction, we have our magnetic field that going into the page, it's values 0.8 Tesla. The route is moving to the right at 7.5 meters per second. The height of this from CD is 50 centimeters and I'm denoting and the length from D to be as l. It's the first first question is defined the induced e m f. So we know that the induced the IMF is gonna be given by our change in flux over a change in time. We can also write our flex as B times a where you don't have to deal with any co sign because the perpendicular of this square region of area and the magnetic field er in the same direction So that coastline is one R b feel this constant so we can pull that out, be changed area over changing time. Our area we see that that is our age Times or L so we can put in e engine edge l And here are height is not changing its only the length from D to be that's changing. So this is B age changing my over changing time. I'm not changing length over changing time is just our velocity that were given. So we have all these numbers and we can see what this value is. Darby is zero point a Tesla. Her height is 50 centimetres, but that in the meters are changing. My cover changing time is 7.5 meters per second. But if you plug all of this in, we get three votes. Now that we have our induced him. If the question is what is the direction of the induced current here, we're gonna need lenses loss. The lenses law tells us that the induced magnetic field establishes to mitigate the change in net magnetic flux. 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 In order, Thio mitigate that change. To reduce that change, the induced Byfield from this little square region must be pointing out of the page. So I'm gonna draw that in red Red will be our abused field. The induced and using our right hand roll here re point our thumb in the direction of the induced the field in the direction that our fingers curl around that vector gives us the direction of the current. So we see that our current was going counterclockwise, counterclockwise induced. Okay, The last part is asking us to find the direction and magnitude of the force required to maintain this uniform motion. So we know that in the setting of a magnetic field, a rod feels a force given by I l B. And one thing I forgot to mention for this part is that we consider a resister in this circuit so resistance of destroying era to that consider a resistance now in the circuit of the 1.5 pumps. So our force on a rod is given by I l B. And we know from homes law that the current is given by our voltage over our resistance. Here are voltages are induced enough. We are in a over What is this thing's times l r. And here are l. The length of the Rod were denoting by age. So let me just correct that really quick. I h b Well, there, there. And now I can see what magnitude this gives us. We have all of these pieces. So we have three over 1.5 are high, is 50 and our magnetic field is point. And this gives us 0.8 news. Now, to think about the direction of this force, let me blow up our bar here. So we know that the current is going up. We know the magnetic field is going into the page, so if we consider a actually make this a little bit bigger. So if we consider a proton in here, this proton, if we use our right hand drawer for the magnetic force, we point our index finger along our velocity. And I'm just gonna you know, that right there would point our index finger along our velocity, the point, our middle finger into the direction of the magnetic field. And our thumb is pointing to the left. So the protons are feeling a force to the left and weaken. Do the same argument for the electrons. That will be the same because the velocity is reversed. But also the charges reversed. So this is telling us the broad is feeling a force from this magnetic field to the left, which means that we must supply force the right and it must be you're a 0.8 noons. If we have this force, it's 0.8 names. The Ron feels no net force, so it can continue at a constant velocity of 7.5 meters per second.

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