How to do this ?

Shouldn’t the answer be 0 since the net enclosed charge is 0 ?

Let us construct a cylinder as shown in the figure, with R (radius of hemisphere) is the radius of the cylinder and 2R is its length. Now you can see that the charge q will be at the center of the cylinder.

Let

S₁ = end surface of the cylinder, φ₁ = flux through S₁

S₂ = half of the curved surface of the cylinder, φ₂ = flux through S₂

S = is the curved surface of the hemisphere and φ = flux through S

Since flux through the central plane is zero, by symmetry we have φ_! + φ₂ = q/2ε_o

We can also see that φ = φ₂ = (q/2ε_o) – φ₁

Since the surface subtends a plane angle of 90^o at the q, the corresponding solid angle can be calculated using the formula , where α is the plane angle subtended.

Thus φ₁ can be calculated as

Finally the required value