Question

A cubic box of side a = 0.819 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) = c1z^2i + c2xyj + c3x^2k, where c1 = 4.16 N/(Cm^2), c2 =

3.62 N/(Cm^2), c3 = 5.48 N/(Cm^2), are constants. What is the electric flux through the top face of the box? (Remember that we define positive flux pointing out of the box.)

Answer 1

Flux through the top face is

Φ = int[int[c3x^2] dx] dy (for 0 < x < a and 0 < y < a)
Φ = int[c3x^3/3] dy

Putting limits for x,
Φ = int[c3(a^3/3)] dy
Φ = c3(a^3/3)y

Puttign limits for y,
Φ = c3(a^3/3)(a^2/2)
Φ = (5.48 N/C-m)(0.819 m)³/3*(0.819)^2/2
Φ = 0.336 N-m²/C