electromagnetism - question on dipole moment of water molecule

I read a line today and don't get it: "Molecules with mirror symmetry like oxygen, nitrogen, carbon dioxide, and carbon tetrachloride have no permanent dipole moments."

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html

So, why molecules with mirror symmetry have no permanent dipole moment?

Answer

While DumpsterDoofus is right, perhaps this explanation might be helpful.

A dipole is an asymmetric separation of charge, like this: $+ -$. A dipole can have many charges. The total charge must be 0. The center of charge for the $+$ charges and the center of charge for the $-$ charges must be at different places.

A dipole can exert electrical forces on nearby charges, because the nearby charge can be closer to the $-$ than the $+$. $F \propto 1/r^3$

Symmetrical arrangements like this are not dipoles. $+--+$. They are similar, in that they can exert forces. But in this example, $F \propto 1/r^4$. The forces from $+$ and $-$ cancel better than for a dipole, but not perfectly.

A CO2 molecule is a line. O=C=O. The O's are a little $-$, and the C is a little $+$. It can have no dipole moment.

H2O is similar, except that it has a V shape. H2O does have a dipole moment.

The O in the middle is slightly $-$. The H's at the ends are two slightly $+$. The center of $+$ charge is half way between the H's.

---

Note that a V shape does have a left-right mirror symmetry. This does not prevent a vertical dipole moment.

Inversion symmetry means that if there is something (perhaps a charge) on one side, then there is another identical something on the opposite side. Example shapes include an X, a snowflake, a circle, and a rectangle.

For a distribution of charges with inversion symmetry, the center of $+$ charge and the center of $-$ charge are always at the center. There can be no dipole.

A starfish and a tetrahedron do not have inversion symmetry. Even so, a tetrahedral molecule (carbon tetrachloride) cannot have a dipole.

In summary, there are many symmetries. Some guarantee no dipole moment. Others do not. It is possible for an asymmetrical charge distribution to have no dipole moment.