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Chapter 28: Problem 5

A \(-4.80 \mu \mathrm{C}\) charge is moving at a constant speed of \(6.80 \times 10^{5} \mathrm{~m} / \mathrm{s}\) in the \(+x\) -direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magneticfield vector it produces at the following points: (a) \(x=0.500 \mathrm{~m}, y=0\) \(z=0 ;\) (b) \(x=0, y=0.500 \mathrm{~m}, z=0 ;\) (c) \(x=0.500 \mathrm{~m}, y=0.500 \mathrm{~m}\) \(z=0 ;(\mathrm{d}) x=0, y=0, z=0.500 \mathrm{~m} ?\)

Short Answer

Expert verified
By following these steps, we can calculate the magnetic field produced by a moving point charge at various points in space. The answers will be different for each point as the position relative to the charge and therefore the vector R change for each point.

Step by step solution

01

Apply the Biot-Savart Law

The Biot-Savart Law allows us to calculate the magnetic field produced by a current. For moving point charges, it is written as: \[ d \mathbf{B} = \dfrac{\mu_{0} q }{4 \pi} \dfrac{v \times \hat{r}}{r^2} \]. Here, \(d\mathbf{B}\) is the magnetic field, \(\mu_{0} = 4 \pi \times 10^{-7} \, Tm/A\) is the permeability of free space, \(q\) is the charge, \(\mathbf{v}\) is the velocity of the charge, \(\hat{r}\) is the unit vector that points from the location of the charge to the location where we are calculating the magnetic field, and \(r\) is the distance from the charge to the location where we are calculating the magnetic field.
02

Calculate the vector R

The vector R is given by \(R = r \hat{r} = \mathbf{r}_{observation} - \mathbf{r}_{charge}\), where \(\mathbf{r}_{observation}\) is the position of the observation point and \(\mathbf{r}_{charge}\) is position of the charge.
03

Determine the velocity vector v

The positive x-direction is defined as the direction of the velocity of the charge, so the velocity vector v is \(v = 6.80 \times 10^{5} m/s \hat{i}\).
04

Calculate B for each point

For each point, calculate the vector R, the cross product of v and R, and use these to calculate the magnetic field vector B.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field
The concept of a magnetic field is fundamental in the study of electromagnetism and is especially pertinent when discussing the forces exerted on moving charges or currents. Imagine a magnetic field as a force field that permeates space where magnetic forces can be felt. This field is created by moving electric charges, such as electrons flowing in a wire, or by magnetic materials like iron magnets.

Understanding the Nature of Magnetic Fields

Unlike electric fields that can emanate from static charges, magnetic fields are exclusively produced by moving charges. This field can be depicted using magnetic field lines, where the direction of the field at any point is tangent to the field line, and the density of these lines indicates the strength of the field.

When a point charge moves, as in our exercise, it generates a magnetic field around it. This field's shape and intensity depend on several factors including the speed of the charge, the direction of its movement, and the presence of other currents or magnets in the vicinity. However, unlike the electric field of a point charge, which spreads out in a sphere, the magnetic field created by a moving point charge is more complex, showing a circular pattern around the path of the charge.
Point Charge
A point charge is an idealized model of a charged particle in which the size of the charge is considered to be so small that it can be represented as a mathematical point. This simplification is valuable as it allows the use of equations to calculate the affects an isolated charge has on its surroundings without worrying about its physical dimensions.

Role of Point Charges in Creating Magnetic Fields

The exercise we are examining involves a moving point charge, which is crucial for the creation of a magnetic field. The charge's motion generates magnetic effects, which can be calculated and visualized at various points in space. According to the Biot-Savart Law, a moving point charge produces a magnetic field that varies inversely with the square of the distance from the charge, and its direction is perpendicular to both the direction of the charge's velocity and the vector pointing from the charge to the point of interest.
Magnetic Field Vector
The magnetic field vector is a mathematical representation of both the magnitude and direction of a magnetic field at a particular point in space. This vector is symbolized by 'B' and is an essential tool in understanding and predicting the behavior of magnetic fields as they interact with charges and currents.

Calculating the Magnetic Field Vector

To calculate the magnetic field vector produced by a moving point charge, one must consider the charge's velocity, the position where the magnetic field is being measured, and the distance between the point charge and that position. As demonstrated in the solution to our exercise, the Biot-Savart Law provides us with the formula to quantify this vector. The calculation involves determining the cross product of the velocity vector of the point charge and the unit vector pointing from the charge to the observation point, illustrating the right-hand rule. It is critical to get the direction of the magnetic field vector right, as it is always perpendicular to both the velocity of the point charge and the line drawn from the point charge to the point of observation.

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Most popular questions from this chapter

A toroidal solenoid with 400 turns of wire and a mean radius of \(6.0 \mathrm{~cm}\) carries a current of 0.25 A. The relative permeability of the core is \(80 .\) (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to the magnetic moments of the atoms in the core?

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 \(28.69 .\)

A solid conductor with radius \(a\) is supported by insulating disks on the axis of a conducting tube with inner radius \(b\) and outer radius \(c\) (Fig. E28.39). The central conductor and tube carry equal currents \(I\) in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. Derive an expression for the magnitude of the magnetic field (a) at points outside the central, solid conductor but inside the tube \((ac)\)

A closed curve encircles several conductors. The line integral \(\oint \overrightarrow{\boldsymbol{B}} \cdot d \overrightarrow{\boldsymbol{\imath}}\) around this curve is \(3.83 \times 10^{-4} \mathrm{~T} \cdot \mathrm{m} .\) (a) What is the net cur- rent in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

A long, straight, solid cylinder, oriented with its axis in the \(z\) -direction, carries a current whose current density is \(\overrightarrow{\boldsymbol{J}}\). The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship $$ \begin{array}{rlr} \overrightarrow{\boldsymbol{J}} & =\frac{2 I_{0}}{\pi a^{2}}\left[1-\left(\frac{r}{a}\right)^{2}\right] \hat{k} & \text { for } r \leq a \\ & =\mathbf{0} \quad & \text { for } r \geq a \end{array} $$ where \(a\) is the radius of the cylinder, \(r\) is the radial distance from the cylinder axis, and \(I_{0}\) is a constant having units of amperes. (a) Show that \(I_{0}\) is the total current passing through the entire cross section of the wire. (b) Using Ampere's law, derive an expression for the magnitude of the magnetic field \(\vec{B}\) in the region \(r \geq a\). (c) Obtain an expression for the current \(I\) contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. (d) Using Ampere's law, derive an expression for the magnitude of the magnetic field \(\vec{B}\) in the region \(r \leq a\). How do your results in parts (b) and (d) compare for \(r=a ?\)
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