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Confirm that the energy in the TEmn mode travels at the

Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths ISBN: 9780321856562 45

Solution for problem 30P Chapter 9

Introduction to Electrodynamics | 4th Edition

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Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths

Introduction to Electrodynamics | 4th Edition

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Problem 30P

Problem 30P

Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the time averaged Poynting vector and the energy density  (use Prob. 9.12 if you wish). Integrate over the cross section of the wave guide to get the energy per unit time and per unit length carried by the wave, and take their ratio.]

Reference prob 9.12

In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ω + δa) and g(r, t) = B cos (k · r − ωt + δb). Show that  where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ω, but they need not have the same amplitude or phase.] For example,

Step-by-Step Solution:

Solution 30P

Step 1 of 5:

In the question, our aim is to show that the energy in the 54CodeCogsEqn.gif mode travels at the group velocity.

We have to start with the time averaged Poynting vector A1CodeCogsEqn.gif and the energy density a2CodeCogsEqn.gif and then integrate them over the cross section of the waveguide to get the energy per unit time and energy per unit length and take their ratio to prove that it will be same as the group velocity.

Step 2 of 5

Chapter 9, Problem 30P is Solved
Step 3 of 5

Textbook: Introduction to Electrodynamics
Edition: 4
Author: David J. Griffiths
ISBN: 9780321856562

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