Flux of an Electric Field — Question 6

Question 6

Translating a cube changes its flux in a quadratic field

A cube of side a spans $x_0\le x\le x_0+a$, $0\le y,z\le a$. In its region the electrostatic field is $\mathbf E=(Cx^2,Dy,0)$, where C and D are constants with appropriate SI units.

1. Calculate its net outward flux using the six faces.

2. Find the change in flux if the cube is translated by distance s along x with its orientation fixed.

Question 6 - Solution

Translating a cube changes its flux in a quadratic field

(a) Method and calculation

$$\Phi_x=Ca^2[(x_0+a)^2-x_0^2]=Ca^3(2x_0+a).$$$$\Phi_y=Da^3,\quad\Phi_z=0,\quad\Phi=a^3[C(2x_0+a)+D].$$

(b) Interpretation and further calculation

$$\Phi(x_0+s)=a^3[C(2x_0+2s+a)+D].$$$$\Delta\Phi=2Ca^3s.$$Flux depends on spatial variation of the field across opposite faces, rather than its absolute value at one face.

Physical check

The constant-gradient y contribution is unchanged by x translation.