there is this problem in section 3.3, of Jackson's classical electrodynamics, which relates to solving Boundary- Value Problems with Azimuthal Symmetry, we have a Conducting sphere with hemispheres at different potentials, I've found the inside and outside potentials

$\phi(r,\theta)$=V \begin{cases}\frac{3}{2}\left(\frac{r}{a}\right) P_{1}(\cos \theta)-\frac{7}{8}\left(\frac{r}{a}\right)^{3} P_{3}(\cos \theta)+\frac{11}{16}\left(\frac{r}{a}\right)^{5} P_{5}(\cos \theta) \cdots, & \text { for } r<a \\ \frac{3}{2}\left(\frac{a}{r}\right)^{2} P_{1}(\cos \theta)-\frac{7}{8}\left(\frac{a}{r}\right)^{4} P_{3}(\cos \theta)+\frac{11}{16}\left(\frac{a}{r}\right)^{6} P_{5}(\cos \theta) \cdots, & \text { for } r>a\end{cases}

And I want to find the surface charge density on the sphere, I don't have any idea on how to approach it.

your help will be very much appreciated.

There is this problem in section 3.3, of Jackson's classical electrodynamics, which relates to solving Boundary- Value Problems with Azimuthal Symmetry, we have a Conducting sphere with hemispheres at different potentials, I've found the inside and outside potentials

$\phi(r,\theta)$=V \begin{cases}\frac{3}{2}\left(\frac{r}{a}\right) P_{1}(\cos \theta)-\frac{7}{8}\left(\frac{r}{a}\right)^{3} P_{3}(\cos \theta)+\frac{11}{16}\left(\frac{r}{a}\right)^{5} P_{5}(\cos \theta) \cdots, & \text { for } r<a \\ \frac{3}{2}\left(\frac{a}{r}\right)^{2} P_{1}(\cos \theta)-\frac{7}{8}\left(\frac{a}{r}\right)^{4} P_{3}(\cos \theta)+\frac{11}{16}\left(\frac{a}{r}\right)^{6} P_{5}(\cos \theta) \cdots, & \text { for } r>a\end{cases}

And I want to find the surface charge density on the sphere, I don't have any idea on how to approach it.

Your help will be very much appreciated.