Radiation E field of a rotor, can't get the result

[fluidistic]

Homework Statement

Two equal and opposite charges are attached to the ends of a rod of length s. The rod rotates counterclockwise at a speed $\omega=ck$. The electric dipole moment of the system at $t=0$ is $\vec p_0=qs\hat x$.
Show that the electric field in the radiation zone is $\vec E_{\text{rad}}(r,\theta, \phi ,t)=\frac{k^2p_0}{4\pi\varepsilon_0}(\cos \theta \hat \theta +i\hat \phi )\frac{e^{i(kr-\omega t + \phi)}}{r}$.
(It's a part of a problem in Zangwill's book).

Homework Equations

$$\vec E _{\text{rad}}=\frac{\mu_0}{4\pi} \frac{\hat r (\hat r \cdot \ddot {\vec p} _{\text{ret}} - \ddot {\vec p}_{\text{ret}})}{r}$$

The Attempt at a Solution

I tried the brute force approach by using the formula I just wrote above, in the relevant equations.
I got that $\vec p (t)=p_0 [\cos (\omega t)\hat x + \sin (\omega t) \hat y]$.
So that $$\ddot {\vec p}_\text{ret}=p_0 \omega ^2 \{ -\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \hat x -\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \hat y \}$$

Converting the Cartesian unit vectors to spherical unit vectors: $\hat x =\sin \theta \cos \phi \hat r + \cos \theta \cos \phi \hat \phi -\sin \phi \hat \phi$. Plug and chug that into the formula for the E field... and I got a non vanishing $\hat r$ component which shouldn't happen.
I reached $$\frac{\mu_0 p_0 \omega ^2}{4\pi r}\{ -\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \cos \phi +\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \sin \phi +\cos \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \cos \phi +\sin \left [ \omega \left ( t-\frac{r}{c} \right ) \right ] \sin \theta \sin \phi \} \hat r$$. I see I can factor out the sine of theta but I don't see how it's going to help me.
I don't seem to reach the desired result.
Any comment is welcome.Edit: Nevermind I see my error. I miscalculated $\hat r \cdot \ddot {\vec p}_{\text{ret}}$. In fact I did $\hat r \cdot \dot {\vec p}_{\text{ret}}$. I now get 0 for the $\hat r$ component as it should. Problem solved!

 

[mark726]

Great job catching your mistake! It's always important to double check your calculations when you're not getting the expected result. Keep up the good work!