Problem 80 - University Physics with Modern Physics

Question

A wide, long, insulating belt has a uniform positive charge per unit area \(\sigma\) on its upper surface. Rollers at each end move the belt to the right at a constant speed \(v\). Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (\(Hint:\) At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem 28.73.)

Step-by-Step Solution

Answer

The magnetic field magnitude is \( \frac{\mu_0 \sigma v}{2} \) and it is directed perpendicular to the belt surface.

Step 1: Understanding Current Density

The moving charged belt can be approximated as an infinite current sheet. The current density, \( J \), due to the moving charges, is given by \( J = \sigma \cdot v \), where \( \sigma \) is the surface charge density and \( v \) is the speed of the belt.

Step 2: Applying Ampère's Law

For an infinite current sheet, the magnetic field near the surface is given by Ampère's Law. The magnetic field \( B \) is determined using the relation \( B = \frac{\mu_0 J}{2} \), where \( \mu_0 \) is the permeability of free space.

Step 3: Substituting Current Density into Ampère's Law

Substitute the current density \( J = \sigma \cdot v \) into the equation for the magnetic field. This results in \( B = \frac{\mu_0 \sigma v}{2} \).

Step 4: Determining the Direction of the Magnetic Field

According to the right-hand rule, if you point your thumb in the direction of the current (movement of the belt), your fingers will curl in the direction of the magnetic field. Therefore, near the surface, the magnetic field will be perpendicular to the belt surface and directed outward.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental concept in electromagnetism. It relates the total magnetic field around a closed loop to the electric current passing through the loop. Mathematically, it is expressed as: \[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\] where \(\mathbf{B}\) is the magnetic field, \(d\mathbf{l}\) is a differential element of the loop, \(\mu_0\) is the permeability of free space, and \(I_{enc}\) is the current enclosed by the loop.
This law helps calculate the magnetic field generated by a current distribution. For an infinite current sheet, as modeled by a moving charged belt, using Ampère's Law simplifies the computation of the magnetic field near its surface. Since the magnetic field is symmetric around the belt, Ampère's Law becomes an effective tool for representing this infinite distribution of current.

Current Density

Current density is a measure of the electric current per unit area of cross-section. It is represented by the symbol \(J\) and is given in units of amperes per square meter (A/m²).
For a moving charged belt, like the one described in the exercise, the current density can be calculated using the formula: \[ J = \sigma \times v \] where \(\sigma\) is the surface charge density (amount of charge per unit area) and \(v\) is the speed of the belt.
Current density is an important concept for understanding how charges in motion (like those on the belt) generate a magnetic field. In this case, with a uniform surface charge density and constant speed, the current is evenly distributed, making it an ideal example of an infinite current sheet.

Surface Charge Density

Surface charge density, represented by the symbol \(\sigma\), describes the amount of electric charge per unit area on a surface. It is measured in coulombs per square meter (C/m²). In electromagnetics, surface charge density contributes to the formation of electric fields, but when in motion, it can also generate magnetic fields.
Given that the belt in the exercise has a uniform positive charge per unit area, its surface charge density can be directly involved in understanding the resultant magnetic field when the belt moves.
In our problem, the surface charge density is the key variable affecting the computation of current density and subsequently, the magnetic field. This is because moving charges translate into currents, which through Ampère's Law, influence the magnetic field around them.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field relative to the direction of the current. To use the right-hand rule:

• Point your thumb in the direction of flow of positive current or movement.
• Curl your fingers around the path of the loop or current sheet.
• The direction your fingers point gives the direction of the magnetic field lines.

In the context of the exercise, the belt moves the charge to the right. Hence, applying the right-hand rule will show that the magnetic field produced by this movement is oriented outward from the surface of the belt.
Understanding this intuitive rule is crucial for visualizing how magnetic fields arise in space around current-carrying conductors, such as our model belt.