Dielectric sphere in constant E-field

[sunrah]

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!

 

[mark726]

Hi there,

I understand your frustration with this problem. It can be quite challenging to work with vector equations and equations involving dielectric materials. Here is a step-by-step guide on how to approach this problem:

1. Start by writing out the equation for the normal component of \vec{D} at the surface:

\vec{D^{ex}_{n}} = \vec{D^{in}_{n}}

2. Substitute in the equations for \vec{D^{ex}_{n}} and \vec{D^{in}_{n}} using the given information:

\epsilon_{0}E^{ex}_{n} = \epsilon_{0}E^{in}_{n} + P

3. Use the definition of \vec{E_{0}} and the given equation for \vec{E^{ex}} to substitute in for E^{ex}_{n}:

\epsilon_{0}(E_{0} - \frac{1}{4\pi\epsilon_{0}}(\frac{P}{r^{3}} - \frac{3P}{r^{5}})) = \epsilon_{0}E^{in}_{n} + P

4. Simplify the equation by multiplying through by r^{3}:

r^{3}\epsilon_{0}E_{0} - P + \frac{3P}{4\pi} = r^{3}\epsilon_{0}E^{in}_{n} + r^{3}P

5. Now, use the definition of \vec{E^{in}} to substitute in for E^{in}_{n}:

r^{3}\epsilon_{0}E_{0} - P + \frac{3P}{4\pi} = r^{3}\epsilon_{0}(E_{0} + \frac{2P}{4\pi\epsilon_{0}})

6. Simplify the equation further by combining like terms and rearranging:

\epsilon_{0}E_{0}r^{3} + P(\frac{3}{4\pi} - r^{3}) = \epsilon_{0}E_{0}r^{3} + P(\frac{2}{4\pi})

7. Finally, divide both sides by r^{3} and simplify to get the desired result:

\epsilon_{0}E_{0} = E_{0} + \frac{2P}{4\pi}

I hope this helps and clarifies the