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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

The law of Biot and Savart in Eq. ( 28.7 ) generalizes to the case of surface currents as $$ \overrightarrow{\boldsymbol{B}}=\frac{\mu_{0}}{4 \pi} \int \frac{\sigma \overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}}}{r^{2}} d a $$ where \(\sigma\) is the local charge density, \(\overrightarrow{\boldsymbol{v}}\) is the local velocity, and \(d a\) is a differential area element. Re-visit Challenge Problem 28.76 and use the above equation as an alternative means to derive the magnetic field at the center of the cylinder. Use the following steps: (a) Write the charge density \(\sigma\). (b) The origin is at the center of the cylinder. What is the vector \(\vec{v}\) that points from the element with coordinates \((x, y, z)=(x, R \cos \phi, R \sin \phi)\) to the origin? (c) What is the velocity \(\overrightarrow{\boldsymbol{v}}\) of the element? (d) What is the vector product \(\overrightarrow{\boldsymbol{v}} \times \hat{\boldsymbol{r}} ?\) (e) An area element on the cylinder may be written as \(d a=R d x d \phi .\) Use this and the previously established information to write the generalized law of Biot and Savart as a double integral. Evaluate the integral to determine the magnetic field \(\vec{B}\) at the center of the cylinder. (f) Is your result consistent with your result in Challenge Problem \(28.76 ?\)

The solenoid is removed from the enclosure and then used in a location where the earth's magnetic field is \(50 \mu \mathrm{T}\) and points horizontally. A sample of bacteria is placed in the center of the solenoid, and the same current is applied that produced a magnetic field of \(150 \mu \mathrm{T}\) in the lab. Describe the field experienced by the bacteria: The field (a) is still \(150 \mu \mathrm{T} ;\) (b) is now \(200 \mu \mathrm{T} ;\) (c) is between 100 and \(200 \mu \mathrm{T},\) depending on how the solenoid is oriented; (d) is between 50 and \(150 \mu \mathrm{T}\), depending on how the solenoid is oriented.

We can estimate the strength of the magnetic field of a refrigerator magnet in the following way: Imagine the magnet as a collection of current-loop magnetic dipoles. (a) Derive the force between two current loops with radius \(R\) and current \(I\) separated by distance \(d \ll R\). Very close to the wire its magnetic field is about the same as for an infinitely long wire, and Eq.( 28.11 ) can be used. (b) Using Eq. ( 28.17 ), express the current \(I\) in terms of the magnetic field at the middle of the loop, and express the radius \(R\) in terms of the area of the loop. In this way, derive an expression for the force \(F\) between two identical current loops separated by a small distance \(d\) in terms of their mutual area \(A\) and center magnetic field \(B\). (c) Rearrange your result to obtain an expression for the magnetic field of a dipole with area \(A\) in terms of the force \(F\) from an identical dipole separated by a small distance \(d\). (d) Now notice that the force it takes to separate one magnet from your refrigerator is nearly the same as the force it takes to separate two magnets stuck together. Estimate that force \(F\). (e) Estimate the area of a refrigerator magnet. (f) Assume that when these magnets are stuck together or to the refrigerator, they are separated by an effective distance \(d=25 \mu \mathrm{m}\). Use the formula derived above to estimate the magnetic field strength of the magnet.

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of \(20.0 \mathrm{~cm}\) and carries a clockwise current of \(12.0 \mathrm{~A}\), as viewed from above, and the outer wire has a diameter of \(30.0 \mathrm{~cm} .\) What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

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.
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