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I don't know how you would orientate the x-y plane, however, the usual way would be to choose the direction of the dipole, i. e. the connection line between both charges, along the z-axis. A reasonable x-y plane with respect to the dipole axis =z-axis would be the symmetry plane perpendicular to the z-axis and symmetric to the charges so that the distance of ...


2

You are right, that the medium is made of quantum systems which have to be treated accordingly. When you consider every individual molecule/atom in your material the induced polarization at that individual molecule/atom is given by the induced dipole moment $d(t)$, which is well defined via $$ d(t) = \langle \psi(t) | \hat d | \psi(t) \rangle, $$ where $\...


1

The dipole is symmetric about its axis, hence the azimuthal symmetry. If the dipole is oriented along $\hat z$ (as it done usually) and one uses cylindrical coordinates with $\hat z$ as the axis of the cylinder, it is clear geometrically that points at constant height $z$ and constant $\rho=\sqrt{x^2+y^2}$, i.e. point in a plane perpendicular to the axis ...


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