Dielectric sphere in constant E-field

sunrah

1. The problem statement, all variables and given/known data
Given a dielectric sphere with relative permittivity = ε in a homogeneous E-field:

[itex]\vec{E_{0}} = E \vec{e_{z}}[/itex].

The E-field causes a homogeneous polarisation (dipole density) of

[itex]\vec{p} = \frac{vec{P}}{V} [/itex] with big P the dipole moment vector. The total electric field outside is:

[itex]\vec{E^{ex}} = \vec{E_{0} - \frac{1}{4 \pi \epsilon_{0}} ( \frac{\vec{P}{r^{3}} - \frac{\vec{3P}{r^{5}}} ) [/itex]

Question: Use the continuous nature of the normal component of [itex]\vec{D}[/itex] at the surface to show that:

[itex]\epsilon_{0} E^{in} = E_{0} + \frac{2P}{4\pi\epsilon_{0}}[/itex]

2. Relevant equations

Using [itex]\vec{D^{ex}_{n}} = \vec{D^{in}_{n}}[/itex] , e.g. normal component of D is continuous at surface
and [itex]\vec{\frac{D^{ex}_{t}}{\epsilon_{0}}} = \vec{D^{in}_{t}}[/itex]


3. The attempt at a solution

I simply don't know what to do I've messed around with this and got nowhere. Please help!

Latex isn't rendering properly - vector arrows are appearing as small boxes but please try and help!!