# Dielectric sphere in constant E-field

## Homework Statement

Given a dielectric sphere with relative permittivity = ε in a homogeneous E-field:

$\vec{E_{0}} = E \vec{e_{z}}$.

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

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

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

Question: Use the continuous nature of the normal component of $\vec{D}$ at the surface to show that:

$\epsilon_{0} E^{in} = E_{0} + \frac{2P}{4\pi\epsilon_{0}}$

## Homework Equations

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

## 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!!

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