A hollow cylindrical box of length 1 m and area of cross -section `25" cm"^(2)` is placed in a three dimensional coordinate system as shown in the figure. The electric field in the region is given by `E=50x hat(i)`, where E is in `"NC"^(-1)` and x is in metre.
Find

net flux through the cylinder,

Explanation

Given, `E=50xhat(i)`
and `DeltaS=25" cm"^(2)=25xx10^(-4)m^(2)`
As the electric field is only along the X-axis, so flux will pass only through the cross- section of cylider Magnitude of electric field at cross-section A,
`E_(A)=50xx1=50" N C"^(-1)`
Magnitude of electric field at cross -section B,
`E_(B)=50xx2=100" NC"^(-1)`
Te corresponding electric fluxes are
`phi_(A)=E_(A)*DeltaS=50xx25xx10^(-4)xxcos180^(@)`
`=-0.125" N-m"^(2)C^(-1)`
`phi_(B)=E_(B)*DeltaS=100xx25xx10^(-4)xxcos0^(@)`
`=0.25"N-m"^(2)C^(-1)`
So, the net flus through the cylinder,
`phi=phi_(A)+phi_(B)=-0.125+0.25=0.125"N-m"^(2)C^(-1)`