Figure 28-56 shows a homopolar generator. which has a solid...

Figure $28-56$ shows a homopolar generator. which has a solid conducting disk as rotor and which is rotated by a motor (not shown). Conducting brushes connect this emf device to a circuit through which the device drives current. The device can produce a greater emf than wire loop rotors because they can spin at a much higher angular speed without rupturing. The disk has radius $R=$ 0.250 $\mathrm{m}$ and rotation frequency $f=4000 \mathrm{Hz}$ and the device is in a uniform magnetic field of magnitude $B=60.0 \mathrm{mT}$ that is perpendicular to the disk. As the disk is rotated, conduction electrons along the conducting path (dashed line) are forced to move through the magnetic ficld.(a) For the indicated rotation, is the magnetic force on those electrons up or down in the figure? (b) Is the magnitude of that force greater at the rim or near the center of the disk? (c) What is the work per unit charge done by that force in moving charge along the radial line, between the rim and the center? (d) What, then, is the emf of the device? (c) If the current is $50.0 \mathrm{A},$ what is the power at which electrical energy is being produced?

Figure $28-56$ shows a homopolar generator. which has a solid conducting disk as rotor and which is rotated by a motor (not
shown). Conducting brushes connect this emf device to a circuit
through which the device drives current. The device can produce a
greater emf than wire loop rotors because they can spin at a much higher angular speed without rupturing. The disk has radius $R=$ 0.250 $\mathrm{m}$ and rotation frequency $f=4000 \mathrm{Hz}$ and the device is in a uniform magnetic field of magnitude $B=60.0 \mathrm{mT}$ that is perpendicular to the disk. As the disk is rotated, conduction electrons along the conducting path (dashed line) are forced to move through the
magnetic ficld.(a) For the indicated rotation, is the magnetic force
on those electrons up or down in the figure? (b) Is the magnitude
of that force greater at the rim or near the center of the disk?
(c) What is the work per unit charge done by that force in moving
charge along the radial line, between the rim and the center?
(d) What, then, is the emf of the device? (c) If the current is $50.0 \mathrm{A},$
what is the power at which electrical energy is being produced?

Key Concepts

Homopolar Generator

A homopolar generator is a type of electrical generator that produces direct current (DC) by rotating a conductive disk in a uniform magnetic field. The design exploits the motion of charges in the disk to induce an electromotive force (emf) between its center and rim, providing a simple, robust means of generating electrical power without the need for commutators that reverse current flow.

Motional EMF

Motional electromotive force arises when a conductor moves through a magnetic field, causing free charges within the conductor to experience a force. This induced force, governed by the Lorentz force law, separates charges, thereby establishing a potential difference along the conductor. In a rotating disk, the effect is integrated along the radial direction, resulting in an emf that depends on the angular speed, magnetic field strength, and the geometry of the disk.

Lorentz Force

The Lorentz force is the force acting on a charged particle moving in a magnetic field and is given by the cross product of the velocity of the charge and the magnetic field. In the context of the homopolar generator, this force acts on electrons moving within the disk, determining both the direction and the magnitude of the charge separation that gives rise to the emf. The right-hand rule is typically applied to determine the direction of the force on the positive charges.

Angular Velocity

Angular velocity is a measure of the rate of rotation of an object and is directly related to the frequency of rotation by the relation ? = 2?f. In rotating systems like the homopolar generator, the tangential velocity of a point on the disk increases with the radial distance, directly affecting the magnitude of the induced emf and the magnetic forces acting on the charges within the disk.

Work per Unit Charge

The work per unit charge in the context of electromagnetic induction represents the energy needed to move a unit charge through the induced electric field created by the motion of the conductor in the magnetic field. This concept is integrated along the path of the charge and is equivalent to the potential difference or emf established across different points in the conductor.

Electrical Power

Electrical power in a circuit is the rate at which energy is converted from one form to another, typically calculated as the product of current and voltage (P = IV). In generators, power conversion involves transforming mechanical energy into electrical energy, where the induced emf drives a current through an external load and the power output is determined by both the magnitude of the current and the voltage produced.

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