I'm tasked with performing the following integral. Specifically, the problem asks the following:

Calculate the potential along the $z$-axis for the case where the upper hemisphere [of a sphere] is at potential $V_0$ and the lower at $-V_0$ by integrating: \begin{equation} \Phi(r,\theta,\phi)= r^{1/2}\partial_r r^{1/2} \int_{-1}^1 d\cos{\theta'} \int_0^{2\pi} \dfrac{d\phi'}{2\pi}\dfrac{af(\theta',\phi')}{\sqrt{r^2+a^2-2ra[\cos{\theta}\cos{\theta'}+\sin{\theta}\sin{\theta'}\cos{(\phi-\phi')}]}} \end{equation}

I can provide more context for the problem if necessary, but I really just need a hand getting started with this integral. I'm outright stuck from the beginning and could use a push in the right direction with integrating the $\phi'$ and $\theta'$ components.

Thank you!

Edit: Since SE broke up the integral on two lines, I'll write it this way to avoid confusion: \begin{equation} \Phi(r,\theta,\phi)= r^{1/2}\partial_r r^{1/2} \int_{-1}^1 d\cos{\theta'} \int_0^{2\pi} \dfrac{d\phi'}{2\pi}\dfrac{af(\theta',\phi')}{\sqrt{r^2+a^2-2ra\cos{\gamma}}} \end{equation} Where $\cos{\gamma} = \cos{\theta}\cos{\theta'}+\sin{\theta}\sin{\theta'}\cos{(\phi-\phi')}$.

  • $\begingroup$ What's $f(\theta',\phi')$? $\endgroup$ – maxmilgram Feb 10 at 21:55
  • $\begingroup$ @maxmilgram Some arbitrary funciton of $f$. $\endgroup$ – Kosta Feb 10 at 21:58

Without explicit knowledge of $f$ you can not calculate the integral. However, observe that your integral still simplifies quite a bit since you are only tasked to find the potential along the z-axis which translates to $\theta=0$ or $\theta=\pi$.


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