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29b. Electromagnetic Induction
Assignment is due at 2:00am on Wednesday, March 14, 2007
Credit for problems submitted late will decrease to 0% after the deadline has passed.
The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help.
The unopened hint bonus is 2% per part.
You are allowed 4 attempts per answer.
Displacement Current and AmpereMaxwell Law
The AmpèreMaxwell Law
Learning Goal:
To show that displacement current is necessary to make Ampère's law consistent for a charging capacitor
Ampère's law relates the line integral of the magnetic field around a closed loop to the total current passing through that loop. This law was extended by Maxwell to include a new type of "current" that is
due to changing electric fields:
.
The first term on the righthand side,
, describes the effects of the usual electric current due to moving charge. In this problem, that current is designated
as usual. The second term,
, is called the
displacement current
; it was recognized as necessary by Maxwell. His motivation was largely to make Ampère's law symmetric with Faraday's law of induction
when the electric fields and magnetic fields are reversed. By calling for the production of a magnetic field due to a change in electric field, this law lays the groundwork for electromagnetic waves in
which a changing magnetic field generates an electric field whose change, in turn, sustains the magnetic field. We will discuss these issues later. (Incidentally, a third type of "current," called
magnetizing current, should also be added to account for the presence of changing magnetic materials, but it will be neglected, as it has been in the equation above.)
The purpose of this problem is to consider a classic illustration of the need for the additional displacement current term in Ampère's law. Consider the problem of finding the magnetic field that loops
around just outside the circular plate of a charging capacitor. The coneshaped surface shown in the figure has a current
passing through it, so Ampère's law indicates a finite value for the field
integral around this loop. However, a slightly different surface bordered by the same loop passes through the center of the capacitor, where there is no current due to moving charge. To get the same loop
integral independent of the surface it must be true that either a current or an increasing electric field that passes through the Ampèrean surface will generate a looping magnetic field around its edge. The
objective of this example is to introduce the displacement current, show how to calculate it, and then to show that the displacement current
is identical to the conduction current
. Assume that the capacitor has plate area
and an electric field
between the plates. Take
to be the permeability of free space and
to be the permittivity of free space.
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 Spring '10
 Donald
 Physics, Magnetic Field, Ampere

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