Two charged hemispheres

Call the larger hemispherical shell $H$ and the smaller shell $K$ . Imagine a second uniformly charged large hemispherical shell $H'$ which, together with $H$ , forms a complete uniformly charged spherical shell with total charge $2Q$ . Let $\mathbf{F}_{HK}$ be the electrostatic force between $H$ and $K$ , and let $\mathbf{F}_{H'K}$ be the electrostatic force between $H'$ and $K$ . Since the electric field inside a hollow uniformly charged hemisphere is zero, we deduce that $\mathbf{F}_{HK} + \mathbf{F}_{H'K} = \mathbf{0}$ .

If we now imagine a second uniformly charged small hemisphere $K'$ , which, together with $K$ , forms a complete uniformly charged spherical shell of charge $2q$ , and let $\mathbf{F}_{HK'}$ be the electrostatic force between $H$ and $K'$ then, by symmetry, $\mathbf{F}_{HK'} = -\mathbf{F}_{H'K} = \mathbf{F}_{HK}$ . Thus $\mathbf{F}_{HK} = \tfrac12\big(\mathbf{F}_{HK} + \mathbf{F}_{HK'}\big)$ , and this last is the electrostatic force between $H$ and the entire smaller spherical sphere. Since the electric field of a uniformly charged spherical shell is, outside that shell, the same as the electric field of a point charge (of the same charge as that on the shell) at the centre of the shell, we deduce that $\mathbf{F}_{HK}$ is equal to $\tfrac12\mathbf{G}_{2q} = \mathbf{G}_q$ , where $\mathbf{G}_u$ is the electrostatic force between $H$ and a point charge $u$ at the centre of the hemispherical shell $H$ .

It is standard bookwork therefore that $\big|\mathbf{F}_{HK}\big| \; = \; |\mathbf{G}_q| \; = \; \boxed{\frac{Qq}{8\pi\varepsilon_0R^2}}$ with the force acting along the line of symmetry of the larger hemispherical shell.