Question

In a region of space the electric field in the `x`-direction and proportional to `x`i.e., `vec(E )=E_(0)xhat(i)`. Consider an imaginary cubical volume of edge a with its sides parallel to the axes of coordinates. The charge inside this volume will be
A. zero
B. `epsilon_(0)E_(0)a^(3)`
C. `(1)/(epsilon_(0))E_(0)a^(3)`
D. `(1)/(6)epsilon_(0)E_(0)a^(3)`

Answer 1

Correct Answer - b
Field at face `ABCD= -E_(0)x_(0)hat(i)`
Flux over the face `ABCD = -E_(0)x_(0)a^(2)`
The negative sign arises as the field is directed into the cube.

Field at face `EFGH= E_(0)(x_(0)+a)hat(i)`
Flux over the face `EFGH= E_(0)(x_(0)+a)a^(2)`
Flux over the other four faces is zero as the field is parallel to the surfaces.
Total flux over the cube `=E_(0)a^(3)=(1)/(epsilon_(0))q`.
Where `q` si the total charge under the cube
`:. q= epsilon_(0)E_(0)a^(3)`.