Why Do Cross-Terms Average to Zero for Incoherent Radiation?

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SUMMARY

The discussion focuses on the mathematical proof that cross-terms average to zero for incoherent radiation, specifically when analyzing the electric field ε(r,t). The participant seeks clarification on the presence of cross-terms and their implications in the context of calculating intensity and energy density. It is established that when summing over different frequencies or applying a Fourier Transform, the cross-terms, which arise from the dot product of ε(r,t), consistently average to zero due to their oscillatory nature.

PREREQUISITES
  • Understanding of electric field representation, specifically ε(r,t)
  • Familiarity with Fourier Transform techniques
  • Knowledge of intensity and energy density calculations in electromagnetic theory
  • Basic grasp of oscillatory functions and their properties
NEXT STEPS
  • Study the properties of Fourier Transforms in the context of electromagnetic fields
  • Explore the derivation of intensity and energy density from electric field expressions
  • Investigate the mathematical treatment of incoherent radiation in electromagnetic theory
  • Learn about the implications of averaging in oscillatory functions and their applications
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Students and researchers in physics, particularly those focusing on electromagnetism and incoherent radiation phenomena, will benefit from this discussion.

davon806
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Homework Statement


Please see the attached,this was extracted from my notes.(Left side is the 1st page).I was trying to prove the sentence underlined (on the right page). Given ε(r,t) , how can I prove that the cross-terms average to 0? Actually I don't really understand why there are cross-terms. Would be much appreciated if someone can elaborate the paragraph on right page in detail !
Q.jpg

Homework Equations

The Attempt at a Solution


To calculate the intensity and energy density, using the formula on left page, I suppose we need to find ε(r,t) ^2 by taking its dot product. However,this would give me a double integral which I don't know how to proceed...
 
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If you sum up and have a bunch of different frequencies (or take an F.T. and get a sum of ## E(\omega)exp^{i \omega t} ## over all frequencies), when you square the expression, the terms involving two frequencies (the cross terms) each time average to zero.
 

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