Problem 87 - University Physics with Modern Physics

Question

A Charged Dielectric Disk. A thin disk of dielectric material with radius \(a\) has a total charge \(+Q\) distributed uniformly over its surface. It rotates \(n\) times per second about an axis perpendicular to the surface of the disk and passing through its center. Find the magnetic field at the center of the disk. (Hint: Divide the disk into concentric rings of infinitesimal width.)

Step-by-Step Solution

Answer

The magnetic field at the center of the disk is \(B = \frac{\mu_0 n Q}{a}\).

Step 1: Understand the Problem

The problem involves calculating the magnetic field at the center of a rotating charged dielectric disk. The disk has a charge +Q uniformly distributed over its surface and rotates at a frequency of n revolutions per second. The approach involves considering the disk as composed of concentric rings to simplify the integration.

Step 2: Divide the Disk into Concentric Rings

Consider an infinitesimally small ring of radius r and thickness dr on the disk. The surface charge density \( \sigma \) is given by \( \sigma = \frac{Q}{\pi a^2} \). The charge on this infinitesimal ring, \( dQ \), is \( \sigma \, \times \, \text{area of the ring} = \sigma (2\pi r \, dr) \).

Step 3: Calculate the Current Due to the Rotating Ring

As the disk rotates, the charges on the ring produce a current. The linear velocity at radius r is \( v = 2\pi r n \). The current \( dI \) due to the rotating charge is \( dI = \frac{dQ}{T} = \frac{dQ}{1/n} = n \, dQ \). Substituting for \( dQ \), we get \( dI = n \, \sigma \, (2\pi r \, dr) \).

Step 4: Use the Biot-Savart Law for Magnetic Field

The magnetic field at the center due to an infinitesimal current element is given by the Biot-Savart Law. For a circular current loop, it gives a field \( dB = \frac{\mu_0 \, dI}{2r} \). Substituting \( dI \), \( dB = \frac{\mu_0 \, n \, \sigma \, (2\pi r \, dr)}{2r} = \mu_0 \, \pi \, n \, \sigma \, dr \).

Step 5: Integrate Over the Entire Disk

Integrate the expression for \( dB \) from \( r = 0 \) to \( r = a \) to find the total magnetic field \( B \) at the center:\[ B = \int_0^a \mu_0 \pi n \sigma \, dr = \mu_0 \pi n \sigma \int_0^a dr = \mu_0 \pi n \sigma a = \mu_0 \pi n \frac{Q}{\pi a^2} a = \frac{\mu_0 n Q}{a}. \]

Step 6: Final Result for the Magnetic Field

The magnetic field at the center of the disk is \( B = \frac{\mu_0 n Q}{a} \). This result shows that the field depends linearly on the number of rotations per second \( n \), the total charge \( Q \), and inversely on the radius \( a \) of the disk.

Key Concepts

Dielectric Disk

A dielectric disk is a type of material that does not conduct electricity but can support electrostatic fields. In our exercise, we have a thin disk of this material with a specified radius and a total positive charge, denoted as \(+Q\), distributed evenly across its surface. This means that the surface of the disk is evenly covered in charged particles. When dealing with charge distributions, we assume that the disk is perfectly planar and uniform, which simplifies the physics involved. The disk in this exercise is rotating around an axis passing through its center, allowing us to study the effects of this motion on the magnetic field produced at the disk's center. By dividing the disk into smaller concentric rings, we can calculate the magnetic contributions from each part of the disk and then sum these to find the total effect.

Surface Charge Density

Surface charge density is a crucial concept when dealing with charged surfaces. It tells us how much charge is present per unit area on the surface. For the dielectric disk, the surface charge density \( \sigma \) is defined as \( \sigma = \frac{Q}{\pi a^2} \), where \(Q\) is the total charge and \(a\) is the radius of the disk. This formula derives from distributing the total charge evenly across the total area of the disk. Understanding surface charge density helps in calculating the charge present on infinitesimally small portions of the disk, like thin rings. By knowing \( \sigma \), you can determine the charge \( dQ \) on an infinitesimally small ring at a radius \( r \) with thickness \( dr \), as \( dQ = \sigma \times (2 \pi r dr) \). This step is fundamental in calculating the current and, subsequently, the magnetic field due to these rotating charges.

Biot-Savart Law

The Biot-Savart Law is a powerful tool in electromagnetism used for calculating the magnetic field generated by a current-carrying conductor. In the context of this problem, we apply it to find the magnetic field at the center of the rotating dielectric disk. It states that the magnetic field \( dB \) due to an infinitesimal current element \( dI \) is proportional to \( \frac{\mu_0 dI}{2r} \) for points along the axis of a circular current loop, where \( \mu_0 \) is the permeability of free space.Applying this law to each infinitesimal ring in the disk gives us the contribution of magnetic field from each ring. We add up (integrate) all these infinitesimal contributions to find the total magnetic field at the disk's center. This process highlights the importance of Biot-Savart Law in calculating fields from distributed sources.

Current Loop

In this exercise, rotating charges on the disk are treated like tiny current loops. Current loops are a classic configuration in which the Biot-Savart Law is often applied. To understand why, think of each rotating ring on the disk as generating a current due to its motion. This is similar to how a current in a wire loop generates a magnetic field.The linear velocity of charges in a ring at radius \( r \) is given by \( v = 2\pi r n \), where \( n \) is the number of rotations per second. This velocity, combined with charge density, allows us to determine the effective current \( dI \) in the ring as \( dI = n \times dQ \). Now, since each concentric ring can be considered an individual current loop, the total magnetic field is a result of integrating the contributions from all such loops across the whole disk. This approach demonstrates a powerful method to analyze magnetic fields resulting from complex charge distributions in motion.