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Chapter 21: Problem 61

Suppose a uniform magnetic field is everywhere perpen dicular to this page. The field points directly upward toward you. A circular path is drawn on the page. Use Amp猫re's law to show that there can be no net current passing through the circular surface.

Short Answer

Expert verified
The net current passing through the circular path is zero by Amp猫re's Law.

Step by step solution

01

Understanding Amp猫re's Law

Amp猫re's Law states that for any closed loop, the line integral of the magnetic field \(B\) along that loop is equal to the permeability of free space \(\mu_0\) times the total current \(I\) passing through that loop: \[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}\] where \(d\vec{l}\) is a differential element of the loop and \(I_{\text{enclosed}}\) is the net current enclosed by the loop.
02

Examining the Magnetic Field Configuration

The magnetic field is uniform and perpendicular to the page and the circular path is drawn on the page. This means that the path is parallel to the plane of the page, and the magnetic field vectors \(\vec{B}\) are perpendicular to any path elements \(d\vec{l}\) of the circular path.
03

Line Integral Calculation

Since the vectors \(\vec{B}\) are perpendicular to \(d\vec{l}\), their dot product is zero along every point of the path: \(\vec{B} \cdot d\vec{l} = B \cdot d\ell \cdot \cos 90^\circ = 0\). As a result, when the total integral around the loop is computed, the result is \(\oint \vec{B} \cdot d\vec{l} = 0\).
04

Conclusion from Amp猫re's Law

According to Amp猫re's Law \(\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}\). Since we calculated \(\oint \vec{B} \cdot d\vec{l} = 0\), it implies that \(\mu_0 I_{\text{enclosed}} = 0\). Therefore, the net current \(I_{\text{enclosed}}\) passing through the circle must be zero.

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Key Concepts

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

Magnetic Field
The magnetic field is an invisible field that exerts a force on charged particles and magnetic materials. In the context of this problem, the magnetic field is uniform and points directly upward from the page, towards you. This orientation indicates that the field lines are perpendicular to the plane of the page. When dealing with magnetic fields, it's important to remember key points:
  • Field lines flow from the north to the south pole outside a magnet.
  • The strength of the field is uniform as stated, meaning it doesn't change in magnitude.
Understanding the direction and uniformity of the magnetic field is crucial when applying Amp猫re's Law. It guides how the field interacts with other elements, like paths and currents, in electromagnetism.
Line Integral
A line integral, in general, is an integral where we integrate a function along a curve. In this context, we specifically deal with the line integral of the magnetic field along a closed path, as described in Amp猫re's Law.
  • The line integral \(\oint \vec{B} \cdot d\vec{l}\) involves summing the magnetic field's influence along the circular path drawn on the page.
  • Every point on the path contributes to this integral based on the magnetic field present there.
  • If the magnetic field is perpendicular to the path (as in this problem), the integral simplifies significantly, resulting in zero.
This simplification is largely due to the dot product \(\vec{B} \cdot d\vec{l} = B \cdot d\ell \cdot \cos 90^\circ = 0\), meaning that the contribution from each differential element of the path is nullified.
Net Current
The concept of net current involves understanding how much total current passes through a defined area, in this case, a circular path. According to Amp猫re's Law, if the line integral of the magnetic field around a loop is zero, then the net current enclosed by that loop must also be zero.
  • Net current, denoted as \(I_{\text{enclosed}}\), is the total current that would be calculated from considering all possible currents passing through the area enclosed by the path.
  • From the solution, since \(\oint \vec{B} \cdot d\vec{l} = 0\), this indicates that no net current passes through our circular path on the page.
This conclusion is pivotal, as it aligns with the principle that without any net current, the interactions, through magnetic fields in this loop, are perfectly balanced.
Circular Path
In this exercise, the circular path is a specific geometric shape drawn on a plane, and it's crucial for applying Amp猫re's Law. The circular path is significant because it forms the loop along which we calculate the line integral.
  • The path is planar, meaning it lies flat on the surface of the page.
  • Because the magnetic field is perpendicular to the page, every point along the circular path experiences the field at a 90-degree angle.
  • This perpendicular interaction simplifies calculations because the magnetic field doesn't contribute to the line integral along the circle.
By understanding the role and orientation of the circular path, we can more easily apply Amp猫re's Law to solve problems related to magnetic fields and currents in specific geometries.

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Most popular questions from this chapter

Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique. Concept Questions (a) A charge moves along the \(+x\) axis and experiences no magnetic force, although there is a magnetic field. What can you conclude about the direction of the magnetic field? (b) A moving charge experiences the maximum possible magnetic force when moving in a magnetic field. What can you conclude about the angle \(\theta\) that the charge's velocity makes with respect to the magnetic field? Explain your answers Problem A particle that has an \(8.2-\mu \mathrm{C}\) charge moves with a velocity of magnitude \(5.0 \times 10^{5} \mathrm{~m} / \mathrm{s}\). When the velocity points along the \(+x\) axis, the particle experiences no magnetic force, although there is a magnetic field present. The maximum possible magnetic force that the charge could experience has a magnitude of \(0.48 \mathrm{~N}\). Find the magnitude and direction of the magnetic field. Note that there are two possible answers for the direction of the field. Make sure that your answers are consistent with your answers to the Concept Questions.

The maximum torque experienced by a coil in a 0.75 -T magnetic field is \(8.4 \times 10^{-4} \mathrm{~N} \cdot \mathrm{m}\). The coil is circular and consists of only one turn. The current in the coil is \(3.7 \mathrm{~A}\). What is the length of the wire from which the coil is made?

The 1200 -turn coil in a dc motor has an area per turn of \(1.1 \times 10^{-2} \mathrm{~m}^{2}\). The design for the motor specifies that the magnitude of the maximum torque is \(5.8 \mathrm{~N} \cdot \mathrm{m}\) when the coil is placed in a 0.20 -T magnetic field. What is the current in the coil?

Consult Interactive Solution \(\underline{21.43} 21.43\) at to see how this problem can be solved. The coil in Figure \(21-22 a\) contains 410 turns and has an area per turn of \(3.1 \times 10^{-3} \mathrm{~m}^{2}\). The magnetic field is \(0.23 \mathrm{~T},\) and the current in the coil is \(0.26 \mathrm{~A} .\) A brake shoe is pressed perpendicularly against the shaft to keep the coil from turning. The coefficient of static friction between the shaft and the brake shoe is \(0.76 .\) The radius of the shaft is \(0.012 \mathrm{~m}\). What is the magnitude of the minimum normal force that the brake shoe exerts on the shaft?

In a television set, electrons are accelerated from rest through a potential difference of 19 \(\mathrm{kV}\). The electrons then pass through a 0.28 - T magnetic field that deflects them to the appropriate spot on the screen. Find the magnitude of the maximum magnetic force that an electron can experience.
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