In the image below, the author works out an example. However, I don't understand how he arrived at $\vec{E}$. In the problem, it says that the charge is uniformly distributed, but somehow he says that the distance from the source point to field point everywhere on the northern hemisphere is at a distance $R$ away. My guess might be that he is using this concept (correct me if I say it wrong): when you have a solid sphere that is uniformly charged, you can pretend that the entirety of the charge is located at its center. Then, you can treat this as a discrete charge which allows you to use a fixed distance to any field point.
However, if this is actually the concept he used I have a problem with it. The southern hemisphere is also charged and this side of the sphere will have a different value for its contribution to the $\vec{E}$ field. In fact, every point on the sphere will have an effect on every other point on the sphere. I know that we are dealing with a continuous distribution of charge here but to help me better visualize the charge I think of the scenario as a HUGE (see: uncountable) collection of point charges. Now I ask the question: How does this point charge affect that point charge? From this perspective, every point on the sphere affects the northern hemisphere.
Could someone help me understand this?
For clarification: I'm not asking about the final solution, I'm only asking about one step in his solution. I don't care about the net force – I only want to know how he arrived at $\vec{E}$.
Best Answer
You are not aswering the question that was asked. You were not asked to compute the force due to the electric field by adding up the force on every bit of the upper hemisphere due to the charges in the lower half. You were instead asked to use the Maxwell stress tensor. The magic of the stress tensor is that you only need to know the total electric field (due to the charges in both hemispheres) on the $surface$ of the part of the object of interest. The total electric field on the curved surface of the upper is the radially outward field of the entire sphere of charge. (you will need to do some work to compute the E field on the flat surface). The Maxwell-stress answer will, of course, be same as the force due to the charges in the lower hemisphere on those in the upper hemisphere. This because the total force on the charges in the upper hemisphere due to the charges only in the upper hemisphere is zero.