# Why does the electric dipole moment of the electron tell us about its sphericity? [duplicate]

There are a bunch of experiments that claim to show that the electron is highly spherical by measuring the electron electric dipole moment. See e.g.:

However, according to the Cartesian expansion of a charge distribution (see http://en.wikipedia.org/wiki/Multipole_expansion), if we assume the electron is made up of two small balls of negative charge $q$ separated by a small distance $2\mathrm{d}z$, and calculate the dipole moment in the centre of the two balls, the dipole moment is zero.

$$(-q)(-\mathrm{d}z)+(-q)(\mathrm{d}z)=0$$

which means even a highly non-spherical shape produces a dipole moment of zero.

So if the dipole moment is measured to be tiny, why does that imply the electron is highly spherical?

Some similar questions on this site:

• What is the mass density distribution of an electron? is a different question because I am not asking about the mass distribution, I am asking about how the dipole moment in a Cartesian expansion tells us about the sphericity of the electron.
• Do electrons have shape? is also not what I am asking; I am asking why it is that the dipole moment can be used to tell you that the electron is spherical when in the example I give of a non-spherical object, the dipole moment is zero and is therefore not a measure of the degree to which the electron is spherical.