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

[blueyellow]

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.

 

[dextercioby]

If it doesn't contain phi, then obviously the derivative of E wrt it is 0...

 

[blueyellow]

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?

 

[dextercioby]

What is B equal to ?

 

[paranormal]

I would first like to clarify that the given equation for E is not complete, as it is missing the radial component. It should be written as E(r, theta, phi, t) = ((A sin theta)/r)(cos(kr - omega t) -(1/kr)sin(kr - omega t)) r-hat + ((A cos theta)/r)(cos(kr - omega t) -(1/kr)sin(kr - omega t)) theta-hat.

Now, to answer the question, let's start by finding the associated magnetic field B using Faraday's law:

From Faraday's law, we know that ∇ x E = -∂B/∂t. Using the given equation for E, we can calculate the curl of E in spherical coordinates:

∇ x E = (1/r^2)(∂(rE_phi)/∂r - ∂E_theta/∂phi) r-hat + (1/r sin theta)(∂E_r/∂phi - ∂(rE_phi)/∂r) theta-hat + (1/r)(∂(rE_theta)/∂r - ∂E_r/∂theta) phi-hat

Since the given equation for E does not contain a phi component, the second term in the above expression becomes zero. Also, since E does not depend on time, the right-hand side of Faraday's law becomes zero. Therefore, we can write:

(1/r^2)(∂(rE_phi)/∂r - ∂E_theta/∂phi) r-hat + (1/r)(∂(rE_theta)/∂r - ∂E_r/∂theta) phi-hat = 0

Solving for B, we get:

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

= (A/c)(cos(kr - omega t) -(1/kr)sin(kr - omega t)) phi-hat

Now, let's move on to showing that E obeys the remaining three of Maxwell's equations:

1. ∇ · E = ρ/ε0

Using the given