Let's assume that I have a sphere of radius $R$ with the spherical coordinates $(r, \theta ,\phi)$ The surface element can be expressed as $dA = R^2 \sin\theta d\theta d\phi$. By integrating over the whole surface, $A = 4\pi R^2$
Now, let's say I consider a point not at the origin but at a distance $d$ from the center. For example, $(0,0,d)$. I am interested in finding the surface density $f(r')$ where $r'$ is the distance from the point to the surface of the sphere.
My expectations are the following:
- If $d = 0$, then the surface density should be such that $f(r'=R)=1$ and $f(r'\neq R)=0$ otherwise.
- If $d = R$, then the surface density should peak at $f(r'=R)$.
My strategy was based on using formulas for the intersection of two spheres. https://mathworld.wolfram.com/Sphere-SphereIntersection.html For a specific $r'$, I can then find the height $R_i$ at which the two circles crosses. This also gives me a condition on the angle of intersection such that $\sin\theta_l = R_i/r'$.
I am struggling with the integration that I should now do to find the surface density.
If I would be interested in the volume, I would expect to integrate the element:
- V = $\int_{R-d}^{R+d}r'^2dr'\int_0^{\theta_l} \sin\theta' d\theta' d\phi$ which simplifies as $2\pi\int_{R-d}^{R+d}r'^2dr'(1-\cos\theta_l) = 2\pi\int_{R-d}^{R+d}r'^2dr'(1-\sqrt{1-R_i^2/r'^2})$.
However, I am not sure on its extension for the surface as both $r'$ and $\theta'$ are varying at the same time. So how to express the surface element in that case?
Another problem is that the intersection of spheres does not seem so suited as $d$ gets closer and closer to zero so a different approach could be more relevent.
I am looking for some ideas on how to pursue. I am also open to a different treatment of the problem so I would welcome new ideas.
At the end, my final goal is to find the surface density as a function of $r'$ for any distance $d$ from the center.
Thank you!