I'm trying to derive the electric field in the centre of a solid hemisphere of radius $ R $ where the charge is distributed uniformly. I have seen different methods involving double/triple integrals but my current knowledge is beyond that and anyway I want to do it the "hard way".
My approach is:
- Calculate the electric field produced by a semi-circle (think of it as a bent rod) of uniform charge. Caring only about the electric field component that doesn't cancel
- Knowing the electric field of a semicircle, calculate the electric field of the hemisphere shell, someway creating that shell from the adding lots of semicircles
- Knowing the electric field of a hemisphere shell, calculate the solid hemisphere integrating the shells from 0 to R
I don't have issues with the first step. The electric field in the axis that doesn't cancel to my calculations and other resources this result:
$$ E_y = \frac{2k\lambda}{R} = \frac{\lambda}{2\pi\epsilon_0R} = \frac{Q}{2\pi^2\epsilon_0R^2} $$
However, I have issues with how to approach the second step. Where now $ Q $ is the charge of the entire hemisphere shell, the electric field component produced by an entire semicircle in the center that won't end up cancelled is:
$$ dE_y = \frac{dQ}{2\pi^2\epsilon_0R^2} \sin(\theta) $$
But I'm sure in how to express $ dQ $ as a function of the angle or even if the approach of adding infinitesimal thin semicircles while rotating them around the center works to create a shell.