The electric field intensity at the centre of a uniformly charged hemispherical shell is $E_0$. Now, two portions of the hemisphere are cut from either side and remaining portion is shown in figure. If $\alpha =\beta =\frac{\pi }{3}$  then electric field intensity at centre due to remaining portion is $\frac{E_0}{n}$, then $n$ is______

The magnitude of electric field intensity due to each part of the hemispherical surface at the center 'O' will be same as all three parts are identical. If we suppose, It is E then the directions of E due to all three are shown in figure below.

At O resulting field is given as

$$\begin{gathered} E+\frac{E}{2}+\frac{E}{2}=E_0 \\ \Rightarrow 2E=E_0 \\ \Rightarrow E=\frac{E_0}{2} \end{gathered}$$