# Surface charge density of Conducting sphere with hemispheres at different potentials

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

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.

• Are you aware of the boundary conditions required for any electric field? Commented Jan 15, 2022 at 12:48
• @Newbie what boundary conditions may I ask? Commented Jan 15, 2022 at 13:09
• en.wikipedia.org/wiki/… Commented Jan 15, 2022 at 13:10
• @Newbie okey thanks, the difference between gradiants of outside and inside potentials equal the surface charge/epslilon, right? Commented Jan 15, 2022 at 13:23
• If you want the two hemispheres to be at different potentials, they will need to be insulated from each other. Commented Jan 15, 2022 at 17:58

If you have found the potential inside and outside the hemispheres, extending Jackson (1.17), $$\vec E_{above} - \vec E_{below}= \frac{\sigma}{\epsilon_0}\hat n \\ \nabla V_{above} - \nabla V_{below}= -\frac{\sigma}{\epsilon_0}\hat n \\$$ Therefore the charge density can be found by, $$\frac{\partial {V_{above}}}{\partial n} - \frac{\partial {V_{below}}}{\partial n} = -\frac{\sigma}{\epsilon_0}$$