Dipole problem with electric fields

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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.