Consider a spherical wave Show that E obeys maxwell's equations

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Homework Help Overview

The discussion revolves around a spherical wave described by an electric field E, with the task of demonstrating that E satisfies Maxwell's equations. The context involves electromagnetic theory, specifically the application of Maxwell's equations to a spherical wave scenario.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to apply Faraday's law to find the associated magnetic field B and to verify that E satisfies the divergence condition of Maxwell's equations. There are questions regarding the differentiation of E with respect to the azimuthal angle phi and the relationship of charge density ro to the terms in E. Another participant is exploring the implications of the Ampere-Maxwell law and questioning the validity of their calculations regarding curl B.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of Maxwell's equations as they relate to the spherical wave. Some guidance has been offered regarding the differentiation process, but there is no consensus on the correct approach to proving compliance with the Ampere-Maxwell law or the divergence condition.

Contextual Notes

Participants are facing challenges due to the lack of dependence of E on the azimuthal angle phi and are questioning how this affects their calculations. There is also uncertainty regarding the relationship between charge density ro and the electric field components.

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Homework Statement



Consider a simple spherical wave, with omega/k=c

E(r, theta, phi, t)=((A sin theta)/r)(cos(kr - omega t) -(1/kr)sin(kr - omega t)) phi-hat

i) Using Faraday's law, find the associated magnetic field B
ii) Show that E obeys the remaining three of Maxwell's equations

The Attempt at a Solution



It's part b I am stuck on. I tried to prove it obeys div E=ro/epsilon0

I tried to do div E in spherical coordinates but I don't know how to when I don't know how to differentiate it with respect to phi because the equation given for E doesn't contain phi. i also don't know how ro is related to any of the terms in the equation for E. Please help.
 
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If it doesn't contain phi, then obviously the derivative of E wrt it is 0...
 
I am now stuck on proving that the wave obey's the Ampere-Maxwell law. I tried to prove that curl B=0. It doesn't. Did I make a mistake in calculating, or is this the wrong approach to take to the question?
 

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