A hollow cylindrical box of length 1 m and area of cross - section `25cm^(2)` is placed in a three dimensional coordinate system as shown in the figure . The electric field in the region is given by `vecE=50xhati` , where E is `NC^(-1)andx` is in metres . Find .
(i) Net flux through the cylinder.
(ii) Charge enclosed by the cylinder.

Explanation

(i) Given :
`vecE=50xhati`
and `DeltaS=25cm^(2)or25xx10^(-4)m^(2)` As the electric field is only along the x -axis , hence , flux will pass only through the cross - section of cylinder.

Magnitude of electrc field at cross - section A,
`E_(A)=50xx1=50N//C`.
Magnitude of electric field at cross - section B,
`E_(B)=50xx2=100N//C`
The corresponding electric fluxes are
`ointphi_(A)=vecE*vecDeltas`
`=50xx25xx10^(-4)xxcos180^(@)`
`=-0.125Nm^(2)//C^(2)`
`ointphi_(B)-vecE*vecDeltas`
`=100xx25xx10^(-4)xxcos0^(@)`
`=0.25Nm^(2)//C^(2)`
So, the net flux through the cylinder,
`ointphi=oint_(A)+ointphi_(B)`
`=-0.125+0.25`
`=0.125Nm^(2)//C^(2)`.
(ii) Using Gauss .s Law
`ointvecE*vecds=(q)/(epsilon_(0))`
`rArr0.125=(q)/(epsilon_(0))`
`rArr0.125=(q)/(8.85xx10^(-12))`
`rArrq=8.85xx10^(-12)xx0.125`
`=1.106xx10^(-12)C`.