One-fourth of a sphere of radius R is removed as shown in figure. An electric field E exists parallel to the xy plane . Find the flux through the remaining curved part.

Options

A. `piR^2R`
B. `sqrt(2)piR^2E`
C. `piR^2E//sqrt(2)`
D. none of these

Answer

C

Explanation

c. `phi_(plain) + phi_(curve) = 0 or phi_(plain) = - phi_(curve)`
`vecA_1 = -(piR^2)/2 hati , vecA_2 = -(piR^2)/2 hatj`
`vecE = E cos 45^@hati + Esin 45^@hatj`
`= E/(sqrt2) hati + E/(sqrt2)hatj`
`phi = vecE * (vecA_1 + vecA_2)`
`= -E/sqrt(2) (piR^2)/2 - E/(sqrt2) (piR^2)/2 = (-piR^2E)/(sqrt2)`
This is the flux entering. So flux is `(piR^2E)/(sqrt2)`.