Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 31: Problem 14

Show that Ampere's Law is not necessarily consistent if the surface through which the flux is to be calculated is a closed surface, but that the Maxwell- Ampere Law always is. (Hence, Maxwell's introduction of his law of induction and the displacement current are not optional; they are logically necessary.) Show also that Faraday's Law of Induction does not suffer from this consistency problem.

Short Answer

Expert verified
Ampere's Law inconsistency occurs when the surface through which the flux is calculated is a closed surface because it does not account for the displacement current in changing electric fields. This can be seen in the charging capacitor example, where Ampere's Law implies there is no magnetic field when the closed loop is considered around the gap between the capacitor plates, despite the wire carrying a current. The introduction of Maxwell-Ampere Law, which includes the displacement current term, resolves this inconsistency.

Step by step solution

01

Ampere's Law

Ampere's Law can be written as: \(\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\) where \(\oint_C \vec{B} \cdot d\vec{l}\) is the magnetic field B integrated over a closed loop C, \(\mu_0\) is the permeability of free space, and \(I_{enc}\) is the total current passing through the surface enclosed by the loop.
02

Consider a charging capacitor

Let's consider the example of a charging capacitor with constant current I and a closed loop C around one of the connecting wires. In this case, Ampere's Law states the following: \(\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I\) However, since the current in the wire is charging the capacitor plates, there is no current passing through the surface enclosed by the loop when the loop is considered around the gap between the capacitor plates. Thus, Ampere's Law implies: \(\oint_C \vec{B} \cdot d\vec{l} = 0\) This creates a contradiction because the wire is carrying a current, but the loop with the gap between the capacitor plates is not carrying any current. Ampere's Law is not consistent in this case.
03

Maxwell-Ampere Law

The Maxwell-Ampere Law, which includes the displacement current term, is given by: \(\oint_C \vec{B} \cdot d\vec{l} = \mu_0 (I_{enc} + \epsilon_0\frac{d\Phi_E}{dt})\) where \(\epsilon_0\frac{d\Phi_E}{dt}\) is the displacement current term, and \(\Phi_E\) is the electric flux.
04

Apply Maxwell-Ampere Law to the charging capacitor

With the charging capacitor scenario, when the loop is around the gap between the capacitor plates, the displacement current term accounts for the changing electric field between the plates. The displacement current is given by: \(\epsilon_0\frac{d\Phi_E}{dt} = I\) Thus, the Maxwell-Ampere Law becomes consistent in this case: \(\oint_C \vec{B} \cdot d\vec{l} = \mu_0(I_{enc} + \epsilon_0\frac{d\Phi_E}{dt}) = \mu_0 I\)
05

Faraday's Law of Induction

Faraday's Law of Induction states that the electromotive force (EMF) induced in a closed loop equals the negative time rate of change of magnetic flux through the loop: \(\oint_C \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}\)
06

Faraday's Law consistency

Faraday's Law is consistent because it is based on the conservation of energy and does not suffer from any inconsistency problems like Ampere's Law. The induced electromotive force in a loop is directly proportional to the rate of change of magnetic flux through that loop. The closed surface for calculating flux is not an issue in this case. In conclusion, we demonstrated that Ampere's Law is not necessarily consistent if the surface through which the flux is calculated is a closed surface. We also showed that Maxwell-Ampere Law addresses this issue, making it always consistent. Furthermore, we verified that Faraday's Law of Induction does not suffer from any consistency problems.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Electric and magnetic fields in many materials can be analyzed using the same relationships as for fields in vacuum, only substituting relative values of the permittivity and the permeability, \(\epsilon=\kappa \epsilon_{0}\) and \(\mu=\kappa_{\mathrm{m}} \mu_{0},\) for their vacuum values, where \(\kappa\) is the dielectric constant and \(\kappa_{\mathrm{m}}\) the relative permeability of the material. Calculate the ratio of the speed of electromagnetic waves in vacuum to their speed in such a material.

Three FM radio stations covering the same geographical area broadcast at frequencies \(91.1,91.3,\) and \(91.5 \mathrm{MHz},\) respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5? Use \(c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\), and calculate the wavelength to an uncertainty of \(1 \mathrm{~mm} .\)

A voltage, \(V\), is applied across a cylindrical conductor of radius \(r\), length \(L\), and resistance \(R\). As a result, a current, \(i\), is flowing through the conductor, which gives rise to a magnetic field, \(B\). The conductor is placed along the \(y\) -axis, and the current is flowing in the positive \(y\) -direction. Assume that the electric field is uniform throughout the conductor. a) Find the magnitude and the direction of the Poynting vector at the surface of the conductor. b) Show that \(\int \vec{S} \cdot d \vec{A}=i^{2} R\)

It is speculated that isolated magnetic "charges" (magnetic monopoles) may exist somewhere in the universe. Which of Maxwell's equations, (1) Gauss's Law for Electric Fields, (2) Gauss's Law for Magnetic Fields, (3) Faraday's Law of Induction, and/or (4) the MaxwellAmpere Law, would be altered by the existence of magnetic monopoles? a) only (2) c) (2) and (3) b) (1) and (2) d) only (3)

An electric field of magnitude \(200.0 \mathrm{~V} / \mathrm{m}\) is directed perpendicular to a circular planar surface with radius \(6.00 \mathrm{~cm}\). If the electric field increases at a rate of \(10.0 \mathrm{~V} /(\mathrm{m} \mathrm{s}),\) determine the magnitude and the direction of the magnetic field at a radial distance \(10.0 \mathrm{~cm}\) away from the center of the circular area.
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Medical Physics

Read Explanation

Classical Mechanics

Read Explanation

Astrophysics

Read Explanation

Mechanics and Materials

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free