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Week 14- Ch 35- Electromagnetic Waves

Due: 11:45pm on Sunday, December 4, 2011

Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy [Switch to Standard Assignment View]

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Electric Field Due to Increasing Flux

Learning Goal: To work through a straightforward application of Faraday's law to find the EMF and the electric field surrounding a region of increasing flux Faraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. This problem is a prototypical example in which an increasing magnetic flux generates a finite line integral of the electric field around a closed loop that surrounds the changing magnetic flux through a surface bounded by that loop. A cylindrical iron rod with cross-sectional area is oriented with its symmetry axis coincident with the z axis of a cylindrical coordinate system as shown. It has a uniform magnetic field inside that varies according to . In other words, the magentic field is always in the positive z direction, and it has no other components. For your convenience, we restate Faraday's law here: , where is the line integral of the electric field, and the magnetic flux is given by , where is the angle between the magnetic field and the local normal to the surface bounded by the closed loop. Direction: The line integral and surface integral reverse their signs if the reference direction of or is reversed. The right-hand rule applies , then the fingers point along . You are free to

here: If the thumb of your right hand is taken along

take the loop anywhere you choose, although usually it makes sense to choose it to lie along the path of the circuit you are considering. Part A Find positive. Hint A.1 Selecting the loop Hint not displayed , the electromotive force (EMF) around a loop that is at distance from the z axis, where

is restricted to the region outside the iron rod as shown. Take the direction shown in the figure as

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Hint A.2

Find the magnetic flux Hint not displayed

Express ANSWER:

in terms of

,

,

,

, and any needed constants such as

,

, and

.

= Correct

Part B Due to the cylindrical symmetry of this problem, the induced electric field the distance Hint B.1 from the z axis, where can depend only on

is restricted to the region outside the iron rod. Find this field.

Calculate the line integral Hint not displayed

Hint B.2

The z and r components of the electric field Hint not displayed

Express

in terms of quantities given in the introduction (and constants), using the unit , , and .

vectors in the cylindrical coordinate system, ANSWER: = Correct

The Ampère-Maxwell 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 right-hand side, . , describes the effects of the usual electric current due to as usual. The second term,

moving charge. In this problem, that current is designated

, 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

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generates an electric field whose...