electrostatics - Why is the radial direction the preferred one in spherical symmetry?

I am learning about electricity and magnetism by watching MIT video lectures.

In the lecture about Gauss's law, while trying to calculate the flux through a sphere with charge in it, the lecturer states that the direction of the electrical field is radial, since it is the only preferred direction that there is (since the problem have spherical symmetry).

But why is this the only preferred direction?

I think that there is also the direction perpendicular to the sphere, but since this is actually a two dimensional subspace of $\mathbb{R}^{3}$ I think that I can rule this out.

Is my conclusion, that since I can't determine any direction (which corresponds to a one dimensional subspace of $\mathbb{R}^{3}$) in any other way, then indeed the redial direction is the only preferred direction?

Answer

You are basically right - I'll just fill in some pedagogical details.

One way to see that there can't be any other direction without worrying about missing possibilities is to suppose for the purpose of contradiction that flux is pointed non-radially. Then use the following definition of spherical symmetry (remember, "spherical symmetry" isn't just a colloquialism - it has a precise meaning):

Any rotation of the system that keeps the center fixed will leave all physically observable quantities unchanged.

Pick a rotation about the axis passing through the center and through the point where you are interested in the flux. Any nonradial component of flux will rotate around this axis, and so you know such a component must be 0.

An analogy would be standing on the Earth's surface and firing a laser into the air. Unless the laser points radially away from (or directly into) the center of the Earth, you could rotate the Earth about the axis passing through your body and the laser would point to a new location.