What is the distribution of surface charge of the inner conductor of this system? [closed]

Question

Given the following system, where the conductors (marked as yellow in the picture) have spherical symmetry. The inner conductor has $\textit{+Q}$ charge and the outer conductor has $\textit{-Q}$ charge. Between both conductors, the space is half filled with a linear, homogeneous dielectric with $\epsilon = 2\epsilon_0$ and air, $\epsilon_0$, in the upper part, as can be seen in the picture.

What is the distribution of surface charge of the inner conductor?

I named the density of surface charge in the upper half of inner sphere, $\sigma_0$, and the density of surface charge in the lower half $\sigma_1$.

I get the following equations: \begin{equation} \left( \sigma_0 + \sigma_1 \right) = \frac{Q}{4\pi a^2} \end{equation}

\begin{equation} \sigma_0 = \sigma_1 \frac{\epsilon_0}{\epsilon} \end{equation}

and so I get \begin{equation} \sigma_1 = \frac{\epsilon_0}{\epsilon_0 + \epsilon_1} \frac{Q}{4 \pi a^2} \hspace{1cm} \sigma_0 = \frac{\epsilon_1}{\epsilon_0 + \epsilon_1} \frac{Q}{4 \pi a^2} \end{equation}

Are these equations right?

Answer 1

For the first equation, you should consider the surface area of a hemisphere as the charge only charge densities are for each hemisphere independently.

The second equation stems from considering the conservation of the parallel components of the electric fields at interfaces.

I would check your results again.