A cubic box of side a = 0.450 m is placed so that its edges are parallel to the

Question

A cubic box of side a = 0.450 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 = 4.10 N/(Cm) 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.) Since E is not constant, you will need to do surface integral for this problem. Remember the definition of electric flux. Only one component of E contributes to the flux out of the top of the box.

Explanation / Answer

here 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 = 4.10 N/(Cm) is a constant. What is the electric flux through the top face of the box? a)

= int[int[E k] dx] dy (for 0 < x < a and 0 < y < a)
= int[int[Ky] dx] dy
= int[Kxy] dy
= int[Kay] dy
= Kay²/2
= Ka³/2
= (4.60 N/C-m)(0.400 m)³/2
= 0.1472 N-m²/C

b)
Since there is no net charge inside the box (not just because the problem says so; div(E) = 0 for this E), the electric flux through the other faces must be:
= -0.1472 N-m²/C

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