The electric field in a region is given by `E=alphaxhati`. Here `alpha` is a constant of proper dimensions. Find
a. the total flux passing throug a cube bounded by the surface `x=l, x=2l, y=0, y=l, z=0, z=l`.
b. the charge contained inside in above cube.

Answer

A, B, C

Explanation

a. Electric field is along positive x x-direction. Therfore,field lines are perpendicular to face `ABCD` and `EFGH`. At all other four faces field lines are tangential. So, net flux passing through these four facess will be zero.
Flux entering at face `ABCD` at this face `x=l`
`:. E=alpha lhati`
`:.` Flux entering the cube from this face
`phi_1=ES=(alphal)(l^2)=alphal^3`
Flux leaving the face `EFGH` At this face `x=2l`
`:. E=2alphalhati`
`:.` Flux coming out of this face
`phi_2=ES=(2alphal)(l^2)`
`=2alphal^3`
`:.` Net flux passing through the cube,
`phi_("net")=phi_2-phi_1=2alphal^3-alphal^3`
`=alphal^3`
b. From Gauss's law,
`phi_("net")=(q_"in")/epsilon_0`
`q_("in")=(phi_("net")(epsilon_0)`
`=alpha epsilon l^3`