Electromagnetisim exam question (can someone please check)

[vorcil]

Homework Statement

A dielectric material has the following properties:
electrical conductivity \sigma = 0
relative permittivity \epsilon_r = 3
relative permeability \mu_r =1

The electric field in the dielectric is given by
\mathbf{E} = E_0 cos(kz-\omega t)\hat{x}

There are no time independent magnetic fields in the dielectric:

i) Write down maxwell's equation in matter in differential form
ii) find an expression for the polarization vector \mathbf{P}
iii) find an expression for the volume density of bound charge
iv) find an expression for the volume density of free charge
v) find an expression for \frac{\parital \mathbf{E}}{\partial \mathbf{t}}
iv) find an expression for \nabla \times \mathbf{E}, and hence deduce an expression for the magnetic field \mathbf{B}, in the dielectric.
vii) find \nabla . \mathbf{B} and explain what this means physically
viii) What is the magnetization vector M in the dielectric?
ix) Find an expression for the phase speed \frac{\omega}{k}

Homework Equations

The Attempt at a Solution

i) Maxwells equations in differential form (THOUGH I'm not sure what they are in matter?)

curl/divergence of both magnetic and electric fields:

\nabla . \mathbf{E} = \frac{\rho}{\epsilon_0} gauss's law
\nabla . \mathbf{B} = 0
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} faraday's law
\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} maxwells fixed ampere's law

the question does say it wants the equations in matter,
so should I be using the auxhillary magnetic field and electric displacement field vectors H and D?

or would the equations in the form I gave be enough?

ii) (writing it up now)

 

[vorcil]

ii)
Find an expression for the polarization vector P:


There are two ways I could think of to approach this question,

using \mathbf{P} =\epsilon_0\chi_e\mathbf{E}

and the electrical displacement route, using \mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}

but in both these methods, I don't know either \chi_e the electrical susceptibility, nor do I know the Displacement vector D,

-

if I were to attempt this in an exam I would have just put down,

\mathbf{P} = \epsilon_0 \chi_e \epsilon_0 cos(kz-\omega t) = \epsilon_0^2 \chi_e cos(kz-\omega t)

 

[vorcil]

iii) Find an expression for the volume density of bound charge:

Using the equation 4.12 from griffiths, \rho_b = -\nabla . \mathbf{P}

If the Polarization vector P, (I had \epsilon_0^2 \chi_e cos(kz-\omega t))is uniform, the volume density of bound charge, \rho_b = 0,

but I'm not sure it is uniform,

does that mean I should take the divergence of the Polarization vector P?
I would use Cartesian coordinates,

(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})(P)?<br /> <br /> I&#039;m not too sure

 

[vorcil]

:( ??