A metal bar with length L, mass m, and resistance R is...

A metal bar with length $L,$ mass $m,$ and resistance $R$ is placed on frictionless metal rails that are inclined at an angle $\phi$ above the horizontal. The rails have negligible resistance. A uniform magnetic field of magnitude $B$ is directed downward as shown in Fig. $P 29.77 .$ The bar is released from rest and slides down the rails. (a) Is the direction of the current induced in the bar from $a$ to $b$ or from $b$ to $a ?$ (b) What is the terminal speed of the bar? (c) What is the induced current in the bar when the terminal speed has been reached? (d) After the terminal speed has been reached, at what rate is electrical energy being converted to thermal energy in the resistance of the bar? (e) After the terminal speed has been reached, at what rate is work being done on the bar by gravity? Compare your answer to that in part (d).

Transcript

00:01 Okay, so we're doing chapter 29, problem 69.

00:04 So in this problem, we have this ramp beautifully drawn here, and is all wired bars, and we have one bar here of length l, and is falling down, and this is all in a magnetic field pointing straight down.

00:23 So as this falls down the ramp, there's a magnetic flux that is induced from the current going through that's completed circuit there.

00:35 Okay.

00:37 So we want to figure out some information about that current.

00:41 So first we want to figure out which direction that current that is going to be induced and this circuit is going to be going.

00:46 Does it flow from a to b in the rod or b to a in the rod? okay.

00:51 So first let's think about this in the frame of lenses law.

00:55 So we have a phi b that is pointing downward with the direction of the magnetic field.

01:02 And the area is decreasing.

01:05 So that means this is decreasing.

01:06 So to oppose that, we must need an induced magnetic field upwards.

01:13 Oh, sorry, induced magnetic field downwards because that is decreasing.

01:20 So if our induced magnetic field is downwards, we use the right -hand rule.

01:24 Let's figure out which way that should go.

01:27 So the right -hand rule should say to produce a downwards thing should be clockwise.

01:32 So that means a to b.

01:35 So current is clockwise, so we flow from a to b.

01:44 Perfect.

01:45 Okay, so for part b says we let this rod fall down such that the only force it feels is the force of gravity and the magnetic force from the induced magnetic field.

01:59 So we want to figure out what the terminal speed is.

02:03 And the terminal speed basically tells us when the magnetic force of the rod is equal to the horizontal component of the normal force, which is normal times tangent phi or mg tenth phi.

02:19 So we want to figure out when the magnetic force is equal to that.

02:23 So what is our expression for magnetic force? well, it's the current times length of the rod times the magnetic field.

02:31 We can relate this even more in terms of emf, emf over the resistance.

02:39 Let's keep going here.

02:41 We can now write the emf using faraday's law as the time derivative in the magnetic flux...

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