A cubic cardboard box of side a = 0.330 m is placed so that its edges are parall

Question

A cubic cardboard box of side a = 0.330 m is placed so that its edges are parallel to the coordinate axes, as shown in the figure. There is NO net electric charge inside the box, but the space in and around the box is filled with a nonuniform electric field of the following form: E(x,y,z) = Kz j + Ky k, where K= 3.60 N/C.m is a constant.

What is the electric flux through the top face of the box? (The top face of the box is the face where z=a. Remember that we define positive flux pointing out of the box.)

Explanation / Answer

flux = integral(E. dA) n^

unit area vector = k E = 3.6z j + 3.6y k

at the face y is variable but z = a

so E = 3.6a j + 3.6y k

taking an elemental area at a distance y from the origin on the surface. i.e. dA = a(dy) k

taking dot product of E and dA; d(flux) = 3.6ay (dy)

total flux will be integral of d(flux) from y=0 to y=a

flux = 3.6a * y^2/2 from y=0 to a givien = 1.8a^2 web/m^2

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