The electric field components are`E_(x)=ax^(1//2),E_(y)=E_(z)=0` in which `a=800NC^(-1)m^(-1//2)`. Calculate a. the flux through the cube, and b. the charge within the cube. Assume that `a=0.1m`

Explanation

Givenn `E_("left end")=alpha x^(1//2)=alpha a^(1//2)` where `x=a`

`E_("Right end")=alpha(2 a)^(1//2)`
Flux `=phi_("Left face")=E_(L)DeltaS cos theta=E_(L)DeltaScos 180^(@)`
through left face
`=-E_(L)DeltaS =-E_(L)a^(2)`
Similarly `phi_("Right face")=E_(R)DeltaS cos 0^(@)=E_(R)a^(2)`
Net electric flux `phi=phi_(L)+phi_(R)`
`=a^(2)(E_(R)-E_(L))`
`a=a^(2)(2^(1//2)alpha a^(1//2)-alpha a^(1//2))`
`=alpha a^(5//2)(sqrt(2)-1)`
but `a=800NC^(-1)m^(-1//2),a=0.1m`
`:.` Electric flux `phi=(800)(0.1)^(5//2)(sqrt(2)-1)`
`=1.05Vm` (or `Nm^(2)C^(-1)`)
b. Net charge `q=epsilon_(0)phi=8.854xx10^(-12)xx1.05C`
`=9.27xx10^(-12)C`