Find the Flux Through a Cube

Bryon

1. The problem statement, all variables and given/known data
A cube, placed in a region of electric field, is oriented such that one of its corners is at the origin and its edges are parallel to the x, y, and z axes as shown to the left. The length of each of its edges is 1.1 m.

(a) What is the flux through the cube if the electric field is given by E = 3.00y?
3*1.1^3 = 3.993

flux = 3.993Nm^2/C

(b) What is the flux through the cube if the electric field is given by E = -4i + (6 +3y)j
flux = ?


2. Relevant equations
flux = E dot A = |E||A|cos(theta)

3. The attempt at a solution

-4(1.1)^3 + (6+3*1.1)*1.1^2 = 5.929Nm^2/C

which isnt correct. I though that you have to find the flux in the x and y directions then just sum them up to get the total flux. Do I need to find the angle between the magnitude of the vectors?

gneill

In (a), the electric field is strictly in the x-direction (parallel to the xy plane), although its magnitude has a y dependency. The important thing, though, is that the direction of the flux lines will be perpendicular to the front and back faces of the cube (where the front and back are assumed to be the faces parallel to the yz plane), and parallel to all other faces.

Things in (b) are a tad more complicated, with the field having distinct x and y (i and j) components. Write the outward pointing unit vectors for the dA area elements for each cube face and take the dot product with the field vector (Cartesian dot product). That will tell you what integrals you'll need to perform to sum up the flux.

Bryon

Would the dot product be like the following:

[-4i + (6 +3y)j] dot y^2

which would just equal (6+3y)(y^2) then just integrate this from 0 to 1.1?

gneill

For the front face it would be [-4i + (6 + 3y)j + 0k] dot [1i + 0j +0k] = -4

For the left side face it would be [-4i + (6 + 3y)j + 0k] dot [0i -1j +0k] = -6 - 3y, where y = 0 for the left side face.

Do the rest accordingly.