* Consider a hemisphere inside a sphere
- The net force on hemisphere is zero,
- \(\Rightarrow\) Force between A and hemisphere is equal and opposite to force between B and hemisphere.
* Now consider outer one hemisphere and inner one sphere.
- Force between hemisphere and ① is same as that between ② and hemisphere.
- In this case the sphere behaves like a point charge at centre
$$F = \int dF \cos\theta = \int \frac{Kq}{R^2} \times \sigma dS \cos\theta$$
$$= \frac{Kq\sigma}{R^2} \times \int dS \cos\theta$$
* \(\int dS \cos\theta\) is the projection of hemisphere’s area on a wall, which is a circle of radius \(R\), So:
$$F = \frac{Kq\sigma}{R^2} \times \pi R^2$$
(Note: The \(R^2\) in the denominator cancels with the \(R^2\) in the numerator)
* For the given case \(q\) and \(Q\) are distributed on hemispheres, so
$$F = Kq\sigma\pi = \frac{1}{4\pi\epsilon_0} \times q \times \frac{Q}{2\pi R^2} \times \pi = \frac{Qq}{8\pi\epsilon_0 R^2}$$