Electromagnetic Waves in Spherical Coordinates

[BOAS]

Hello,

I am trying to find the magnetic field that accompanies a time dependent periodic electric field from Faraday's law. The question states that we should 'set to zero' a time dependent component of the magnetic field which is not determined by Faraday's law. I don't understand what is meant by this.

1. Homework Statement

Consider the following periodically time-dependent electric field in free space, which describes a certain kind of wave.

##\vec E (r, \theta, \phi, t) = A \frac{\sin \theta}{r} \cos(kr - \omega t) \hat \phi##, where ##\omega = ck##

(a) Show that, for r > 0, E~ satisfies Gauss’s law with no charge density.
From Faraday’s law, find the magnetic field. Ignore (set to zero) a time dependent part of the B~ -field not determined by Faraday’s law.
(b) Compute the Poynting vector ##\vec S##.
(c) Calculate ##\bar S##, the average of ##\vec S## over a period ##T = 2π/ω ##.
(d) Find the flux of ##S##through a spherical surface of radius ##r## to determine the total power radiated.

Homework Equations

The Attempt at a Solution

[/B]
Part (a) is obvious because the ##\hat \phi## component has no dependence on ##\phi##

part(b)

Given ##\vec E (r, \theta, \phi, t) = A \frac{\sin \theta}{r} \cos(kr - \omega t) \hat \phi##.

I use Faraday's law ##\vec \nabla \times \vec E = - \frac{\partial \vec B}{\partial t}## and the expression of the curl in spherical polar coordinates to find that;

##\vec \nabla \times \vec E = \frac{2A \cos \theta}{r^2} \cos(kr - \omega t) \hat r + kA \sin \theta \sin(kr - \omega t) \hat \theta##.

Integrating with respect to time to find ##\vec B## yields;

##\vec B = - [\frac{2A \cos \theta}{r^2} \hat r \int \cos(kr - \omega t)dt + kA \sin \theta \hat \theta \int \sin(kr - \omega t)dt]##

##\vec B = \frac{2A \cos \theta}{r^2 \omega} \sin(kr - \omega t) \hat r - \frac{kA \sin \theta}{\omega} \cos(kr - \omega t) \hat \theta + C##

I think that this is the magnetic field, but I haven't used the piece of information given in the question about 'setting the time dependent component to zero'.

What does that piece of information mean here?

Thank you.

 

[nrqed]

Hello,

I am trying to find the magnetic field that accompanies a time dependent periodic electric field from Faraday's law. The question states that we should 'set to zero' a time dependent component of the magnetic field which is not determined by Faraday's law. I don't understand what is meant by this.

1. Homework Statement

Consider the following periodically time-dependent electric field in free space, which describes a certain kind of wave.

##\vec E (r, \theta, \phi, t) = A \frac{\sin \theta}{r} \cos(kr - \omega t) \hat \phi##, where ##\omega = ck##

(a) Show that, for r > 0, E~ satisfies Gauss’s law with no charge density.
From Faraday’s law, find the magnetic field. Ignore (set to zero) a time dependent part of the B~ -field not determined by Faraday’s law.
(b) Compute the Poynting vector ##\vec S##.
(c) Calculate ##\bar S##, the average of ##\vec S## over a period ##T = 2π/ω ##.
(d) Find the flux of ##S##through a spherical surface of radius ##r## to determine the total power radiated.

Homework Equations

The Attempt at a Solution

[/B]
Part (a) is obvious because the ##\hat \phi## component has no dependence on ##\phi##

part(b)

Given ##\vec E (r, \theta, \phi, t) = A \frac{\sin \theta}{r} \cos(kr - \omega t) \hat \phi##.

I use Faraday's law ##\vec \nabla \times \vec E = - \frac{\partial \vec B}{\partial t}## and the expression of the curl in spherical polar coordinates to find that;

##\vec \nabla \times \vec E = \frac{2A \cos \theta}{r^2} \cos(kr - \omega t) \hat r + kA \sin \theta \sin(kr - \omega t) \hat \theta##.

Integrating with respect to time to find ##\vec B## yields;

##\vec B = - [\frac{2A \cos \theta}{r^2} \hat r \int \cos(kr - \omega t)dt + kA \sin \theta \hat \theta \int \sin(kr - \omega t)dt]##

##\vec B = \frac{2A \cos \theta}{r^2 \omega} \sin(kr - \omega t) \hat r - \frac{kA \sin \theta}{\omega} \cos(kr - \omega t) \hat \theta + C##

I think that this is the magnetic field, but I haven't used the piece of information given in the question about 'setting the time dependent component to zero'.

What does that piece of information mean here?

Thank you.

Are you sure they did not mean to say "set to zero a time INdependent part of B?? That would make more sense (note that you did that by not including a constant of integration)