Electric Dipoles and Spherical Coordinates

I've become confused over the use of spherical coordinates when working with dipole moments. It would probably be best o use an example to show where I'm confused.

If we have a pure dipole, with a dipole moment pointing angle $\theta$ from the vertical axis (which we'll call $x$), it's dipole moment is given by the vector $\vec{p} = pcos( \theta) \hat{x} + psin( \theta) \hat{y} + 0 \hat{z}$ in Cartesian coordinates, but my book claims that it is given by (in spherical coordinates) $\vec{p} = pcos( \theta) \hat{r} + psin( \theta) \hat{\theta}$ .

How is that possible? If you want to change from Cartesian to spherical, you use the conversation equations, which equal $\vec{p} = p \hat{r} + \theta \hat{ \theta} + 0 \hat{ \phi}$. The rest of the problem asks to find the torque on this dipole moment due to a conductor a distance away (method of images).

I get the process, but the coordinates are really causing problems for me.

I'm not asking for help solving the problem, I'm mostly confused with the coordinates, I think my book may have made a mistake.

The dipole is presumably, not at the origin! Although they have vaguely double used $\theta$; this is how the situation should look:

• I see, thanks for the graph, it finally clicked, I always get the units jumbled in my head. Jul 24, 2013 at 7:13
• @Astrum I'm glad I could help.
– Ali
Jul 24, 2013 at 7:15
• The dipole could be at the origin as well. This will not change the result. Jul 24, 2013 at 10:25
• @Trimok it definitely will. At the origin $\hat r$ and $\hat \theta$ are not well defined!
– Ali
Jul 24, 2013 at 10:31
• @Ali : $r \hat r$ is not necessarily the position of the origin of the dipole. it could be an (external) new choice of coordinates, independant of the position of the origin of the dipole. Jul 24, 2013 at 11:03

There's nothing wrong or right in choosing Cartesian over Spherical coordinates or vice-versa. Its just that in some cases, the math in one can be easier to handle over the other.

Maybe you are getting confused in the $\theta$ chosen for both notations(they are different).

The $\theta$ you chose for the Cartesian notation, is the angle $\vec p$ makes with $\hat x$.

The $\theta$ they have chosen is the angle $\vec p$ makes with $\hat r$. Maybe that's why your conversion equation doesn't work.

More on Spherical coordinate notation here.