Electromagnetisim exam question (can someone please check)

1. The problem statement, all variables and given/known data
A dielectric material has the following properties:
electrical conductivity [tex]\sigma = 0 [/tex]
relative permittivity [tex]\epsilon_r = 3 [/tex]
relative permeability [tex]\mu_r =1 [/tex]

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

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 [tex] \mathbf{P} [/tex]
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 [tex] \frac{\parital \mathbf{E}}{\partial \mathbf{t}} [/tex]
iv) find an expression for [tex] \nabla \times \mathbf{E} [/tex], and hence deduce an expression for the magnetic field [tex] \mathbf{B} [/tex], in the dielectric.
vii) find [tex] \nabla . \mathbf{B} [/tex] and explain what this means physically
viii) What is the magnetization vector M in the dielectric?
ix) Find an expression for the phase speed [tex] \frac{\omega}{k} [/tex]

2. Relevant equations

3. 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:

[tex] \nabla . \mathbf{E} = \frac{\rho}{\epsilon_0} [/tex] gauss's law
[tex] \nabla . \mathbf{B} = 0 [/tex]
[tex] \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} [/tex] faraday's law
[tex] \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}[/tex] 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)
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Find an expression for the polarization vector P:

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

using [tex] \mathbf{P} =\epsilon_0\chi_e\mathbf{E} [/tex]

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

but in both these methods, I don't know either [tex] \chi_e [/tex] 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,

[tex] \mathbf{P} = \epsilon_0 \chi_e \epsilon_0 cos(kz-\omega t) = \epsilon_0^2 \chi_e cos(kz-\omega t) [/tex]
iii) Find an expression for the volume density of bound charge:

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

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

[tex] (\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})(P)???

I'm not too sure
:( ??

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