Problem 76 - Physics for Scientists and Engineers with Modern Physics

Question

A circular metal disk of radius \(R\) rotates with angular velocity \(\omega\) about an axis through its center perpendicular to its face. The disk rotates in a uniform magnetic field \(B\) whose direction is parallel to the rotation axis. Determine the emf induced between the center and the edges.

Step-by-Step Solution

Answer

The induced emf is \( \varepsilon = \frac{1}{2} B \omega R^2 \).

Step 1: Identify the Scenario

We are dealing with a circular metal disk of radius \( R \) rotating with angular velocity \( \omega \) in a uniform magnetic field \( B \), which is parallel to the axis. The key is to find the induced emf from the center to the edge due to this rotation.

Step 2: Understand Relevant Formula

The formula for the induced emf \( \varepsilon \) in a rotating disk in a magnetic field is given by: \( \varepsilon = \frac{1}{2} B \omega R^2 \). This result comes from the integration of the magnetic force on the charges in the disk, considering the disk's rotation.

Step 3: Apply the Formula

Substitute the given quantities into the formula: the magnetic field \( B \), angular velocity \( \omega \), and radius \( R \) into \( \varepsilon = \frac{1}{2} B \omega R^2 \).

Step 4: Calculate the Induced EMF

Perform the calculation using the known values. Multiply \( B \), \( \omega \), and \( R^2 \), and then take half. Thus, the expression for the induced emf becomes \( \varepsilon = \frac{1}{2} B \omega R^2 \).

Key Concepts

Rotating Disk in Magnetic Field

When a circular metal disk is placed in a magnetic field and set to rotate, fascinating phenomena occur. Here, the key scenario involves a disk rotating about its central axis. The axis is perpendicular to its face and is parallel to a uniform magnetic field. This configuration allows the magnetic field lines to interact effectively with the rotating disk, leading to interesting results in electromagnetic phenomena.

As the disk spins, each part of it moves through the magnetic field lines. This motion relative to the magnetic field is crucial, as it's the source of induced electromotive force (emf) in the disk. The magnetic field can be imagined as being constant and uniform, simplifying the calculations and helping us clearly understand the physics happening in the setup.

• The disk rotates about its center, interacting with a magnetic field parallel to the axis.
• Motion across magnetic field lines is key to inducing emf.

Overall, understanding this setup is critical when analyzing situations involving magnetic induction in rotating bodies.

Induced EMF

The primary result of having a rotating disk in a magnetic field is the induced electromotive force (emf). This phenomenon occurs due to the motion of the conducting disk in the magnetic field. As parts of the disk cut across the magnetic field lines, a force is exerted on charge carriers within the metal of the disk. This force, called the Lorentz force, pushes charges and causes them to move, creating a potential difference.

In this context, the induced emf can be calculated using the formula: \( \varepsilon = \frac{1}{2} B \omega R^2 \). This formula shows that the induced emf depends on several factors:

• **Magnetic Field** \( B \): A stronger field results in a greater induced emf.
• **Angular Velocity** \( \omega \): Faster rotation increases the emf.
• **Radius** \( R \): Larger disks see a greater effect, evidenced by the \( R^2 \) term in the formula.

Hence, the induced emf represents how effectively the disk generates an electrical potential due to its rotation through magnetic field lines.

Angular Velocity

Angular velocity, denoted by \( \omega \), describes the speed of rotation of an object around an axis. It's a vector quantity, indicating not only how fast something spins but also the direction of rotation. For the rotating disk in the magnetic field, angular velocity is a critical factor in determining the magnitude of the induced emf.

The relationship between angular velocity and induced emf is direct and proportional. This means that as angular velocity increases, the induced emf also increases linearly, provided that other factors like the magnetic field strength and the disk's radius remain constant.

Understanding this relationship helps us predict how changes in rotational speed affect the electromagnetic characteristics of the system. In practical terms, if the disk spins faster, it encounters more field lines per unit time, thus:

• **Higher \( \omega \)** — More significant interaction with magnetic fields, leading to larger emf.
• **Lower \( \omega \)** — Reduced interaction, resulting in a smaller emf level.

Therefore, controlling angular velocity allows control over the induced emf, which is invaluable in engineering and technology applications.