Intensity of quadrupole radiation as a function of speed

[dingo_d]

Homework Statement

Proton of mass m and charge e is moving in arbitrary direction in external homogeneous constant electric field E. Show the intensity I of quadrupole radiation as a function of proton speed and magnitude of electric field.

The Attempt at a Solution

I have similar problem, but for magnetic dipole radiation. And there it's not hard to solve it because intensity is given by:

$$I=\frac{2}{3c^3}(\ddot{m})^2$$, and $$\vec{m}=\frac{1}{2c}e\vec{r}\times\vec{v}$$.

But here I have quadrupole radiation. Do I need to find the quadrupole moment? It's a tensor, right?

I'm kinda stuck.

I have the equation of motion $$m\ddot{\vec{r}}=e\vec{E}$$, but other than that I'm stuck :\

 

[antonniozb]

Hello I'am studying radiation in the far field approximation (r'<<\lambda<<r) and it's possible to deduce the intensity of radiation (by averaging over the unit sphere, i.e integrating over the solid angle).

$$
I=(2/3c^{3})(\dott{m}^{2}+\dott{P}^{2})+\frac{1}{180c^{5}}(\sum_{i,j}Q_{i,j}^{2})
$$

where $Q_{i,j}$ is the dipole moment, defined by:

$$
Q_{i,j}=\int[3x_{i}x_{j}-\delta_{ij}r`^ {2}]\rho(r',t)dv'
$$

the charge density for point charges is celebrated so the deduction is trivial
by using the delta function fixing the charge distribution position. My doubt in the formula relies in the fact that I'am not sure if in the sum over (ij) we have to consider $i\neq j$ or qe repeat twice ij when they are different.