Dipole problem with electric fields

[bubblewrap]

Dipole problem (which is solved through mirror imaging) has been troubling me with its solution. I understand everything except how the dipole moment's coordinates came to be, since when converted into x-y axis, its doesn't make sense. (problem 4.6)

The screenshot contains the solution which says that dipole moment p can be expressed into the terms shown (at the end of step 2, right before step 3

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[Charles Link]

It helps to start with a good source. Try: http://phys.columbia.edu/~nicolis/Dipole_electric_field.pdf $\\$ Meanwhile $\vec{p}_{image}=-p \cos{\theta} \, \hat{z} +p \sin{\theta} \, \hat{x}$ where we will assume the dipole has $\phi=0$ for the angle which it points.$\\$ Using $\vec{r}=2 z \, \hat{z}$, $\vec{ E}$ (at location $(0,0, 2z)$ from $\vec{p}_{image}$), can be computed from $\vec{p}_{image}$. $\\$ Meanwhile, $\vec{p}=p \cos{\theta} \, \hat{z}+ \sin{\theta} \, \hat{x}$ should make it an easy algebraic exercise to compute the torque.$\\$ I do think your book made a mistake in computing $\vec{p}$. $\\$ I got an expression for electric field $E$ and torque $\vec{N}=\vec{p} \times \vec{E}$. I'd be happy to see how your result compares to mine.