A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad / s about an axis throug

step 1 Okay, so we're doing chapter 29, problem 59. So in this problem, we have a rod that is on axis h

A slender rod, 0.240 m long, rotates with an angular speed...

A slender rod, $0.240 \mathrm{~m}$ long, rotates with an angular speed of $8.80 \mathrm{rad} / \mathrm{s}$ about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of $0.650 \mathrm{~T}$. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at $8.80 \mathrm{rad} / \mathrm{s}$ about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

Key Concepts

Potential Difference in Conductors

The potential difference or voltage between two points in a moving conductor arises due to the separation of charge caused by the induced motional emf. In scenarios involving rotating conductors, calculating this potential difference involves integrating the induced electric field along the path between the two points, taking into account the distribution of speeds along the length of the conductor.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that a change in magnetic flux through a circuit or a conductor induces an electromotive force (emf). Although here the emphasis is on the motional aspect of induction rather than a changing magnetic field, the principle underlies the process of converting mechanical motion (rotation) into electrical energy, as seen in the induced emf along the rotating rod.

Motional Electromotive Force (EMF)

Motional EMF is generated when a conductor moves through a magnetic field, causing free charges within the conductor to experience a magnetic Lorentz force. This force separates charges along the length of the conductor and produces a potential difference. In a rotating rod, the induced emf can be obtained by integrating the contributions of motional emf along its length.

Rotational Motion and Angular Velocity

In a rotating system, every point of a rigid body moves with a linear speed that is proportional to its distance from the axis of rotation, using the relationship v = ?r. This concept is crucial for determining how different segments of a rotating rod contribute differently to the induced emf since points farther from the rotation axis have higher linear speeds.

Lorentz Force

The Lorentz force, defined as F = q(v × B), is the fundamental force acting on moving charges in a magnetic field. It explains the mechanism behind motional EMF: as the rod rotates, each infinitesimal segment with its associated charge experiences this force, resulting in a separation of charges and the creation of a measurable potential difference.

Transcript

00:01 Okay, so we're doing chapter 29, problem 59.

00:05 So in this problem, we have a rod that is on axis here.

00:09 So it's free to rotate upwards.

00:12 And it is actually in motion upwards right now.

00:17 And the rod is a length l.

00:21 And it's all in a magnetic field that is out of the page.

00:32 Okay, cool.

00:35 So what we want to do is find with the induced emf is on this rod.

00:41 So to do this, we're going to have to use this expression here, which is that the de is b cross b dotted into dl of the rod.

00:55 Okay, so this is pretty simple because we just do v times b, the r, because we're all, we're going to just all work in the radial direction here.

01:05 And we can also know that only, sorry, that v is the same as a omega times r.

01:12 So this is b omega r d r.

01:16 Cool.

01:17 So now if we're looking for the total e -map, we integrate the differential, which is the same as integrating to r from zero to l over the whole distance of the rod.

01:33 Omega times b turns r d r.

01:37 So if we integrate this, we get one half omega b r squared, evaluate zero that and that becomes one half omega b l squared we plug in our values for this they tell us some of the stuff they tell us omega is 8 .8 radiums per second they tell us b is 0 .65 teslas and l is 0 .24 meters so we can plug all that in and we see this comes out to an induced emf of 0 .165 bolts.

02:26 So now part b is asking what's the potential difference between the ends of the rod.

02:31 So this is just a rod.

02:33 There's no circuit here...

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