Scale invariance symmetry as a simple argument in an electrostatics problem

In the comments to this post, it was hinted that proving that the force acting on a charge at a vertical distance from a uniformly charged plane is independent of that distance can be done by recalling the scale invariance symmetry. Can someone explain that to me?

Answer

Here's Coulomb's Law:

If you scale everything by $\lambda$, you get $\frac{1}{\lambda^2}$ in the denominator, but you must also introduce a Jacobian for the integral.

For a volume charge, the Jacobian is $\lambda^3$, so you're on the surface of a ball of charge and you make the ball bigger (with the same charge density), the E-field increases proportionately.

For a surface charge, though, the Jacobian is $\lambda^2$, which cancels the $\frac{1}{\lambda^2}$ in the denominator of Coulomb's law. Thus, a 2-d distribution of charge is "scale invariant". Since scaling an infinite sheet leaves it unchanged, scaling changes only the distance from the sheet, and we see that the E-field is independent of distance.