A point charge +Q is place just outside an imaginary hemispherical surface of radius R as shown in the figure. Which of the following statements is/are correct ?

Options

A. The electric flux passing through the curved surface of the hemisphere is `- (Q)/( 2 epsi_(0)) ( 1-(1)/(sqrt(2))`
B. The component of the electric field normal to the flat surface is constant over the surface
C. Total flux through the curved and the flat surfaces is `(Q_(0))/(epsi_(0))`
D. The circumference of the flat surface is an equipotential

Answer

A, D

Explanation

Since the charge lies outside the hemi-sphere, net flux passing through the sphere is zero.
`phi_("curved surface")+phi_("disc")=0`
Option (C) is incorrect
`phi_("curved surface")=-phi_("disc")`
`costheta=R/(sqrt(R^(2)+r^(2))`, `phi_("disc")=intvecE.dvecA`

`=int(1/(4piepsi_(0))Q/((R^(2)+r^(2)))xx2pirdrxxcostheta)=(Q.2pi)/(4piepsi_(0))int(rdr)/(R^(2)+r^(2))xxR/((R^(2)+r^(2))^(1//2))`
`=(QR)/(2epsi_(0))int_(0)^(R)(rdr)/((R^(2)+r^(2))^(3//2))=(QR)/(2epsi_(0))[1/2((R^(2)+r^(2))^(-1//2))/(-1//2)]_(0)^(R)=(QR)/(2epsi_(0))[1/R-1/(sqrt(R^(2)+R^(2)))]=Q/(2epsi_(0))[1-1/(sqrt2)]`
`thereforephi_("curved surface")=-Q/(2epsi_(0))[1-1/(sqrt2)]` Option (A) is correct
Potential at any point on the circumference of the flat surface is `1/(4piepsi_(0))Q/(sqrt(R^(2)+R^(2)))=Q/(4piepsi_(0)(sqrt2R))`
Hence it is equipotential
Option (D) is correct
`E=1/(4piepsi_(0))Q/((R//costheta)^(2))=(theta)/(4piepsi_(0)R^(2))cos^(2)theta`
`E_("normal")=Ecostheta=Q/(4piepsi_(0)R^(2))cos^(3)theta`

Which is not constant. Option (B) is in correct