Cube Electric Field: Are Intersections Uniformly Distributed?

[bocobuff]

Homework Statement

A positive charge is located at the center of a cube.
Are the intersections of the field lines with a side of the box uniformly distributed across that side? Explain

The Attempt at a Solution

I'm trying to picture this in my head and I'm getting stuck. I know the field lines in the corners are further away from the charge than the lines in the center of the cube face. But I can't determine if more lines would be coming into the corner of the face than the center, which would be perpendicular to the charge. Also, if the field lines did all intersect uniformly, wouldn't it be a sphere rather than a cube? I keep trying to visualize it like a golf ball in a kleenex box with lines coming out of the all dimples but I think I'm really over-thinking this one. Help if you can. Thanks.

 

[LowlyPion]

The corners are how much farther away than the middle of a face?

a/2 as opposed to √ 3*a/2

If the density of the lines is proportional to strength ...

 

[bocobuff]

and the greater the distance, the less the strength?

 

[Delphi51]

I've drawn a circle inscribed in a square, but I'm thinking of a sphere inscribed in a cube. The field line density around the sphere is uniform, N/Ss where N is the number of lines and Ss the surface area of the sphere.

The same N lines go through the enclosing cube so its density averages N/Sc.
But Sc > Ss so N/Sc < N/Ss. Less dense on the cube, on average.
However, at or near those points where the sphere touches the cube, the density is the same as on the cube - greater than the average on the cube.


In one corner you have N/4 lines coming out of the quarter sphere and then going through a quarter of the cube. The surface area of the quarter cube > surface area of the quarter sphere. Therefore the density of the lines is less on the cube than on the sphere. Except at or near the points where the sphere touches the cube and the line density of the cube is equal to that on the sphere.