According to my textbook, the correct formula for the electric field produced by a moving charge is $$\vec E=-\frac{q}{4\pi\epsilon_0}\left[\frac{\hat e_{r'}}{r^{'2}}+\frac{r'}{c}\frac{d}{dt}\left(\frac{\hat e_{r'}}{r^{'2}}\right)+\dfrac{1}{c^2}\dfrac{d^2\hat e_{r'}}{dt^2}\right]$$ where $r'$ is the retarded position and $\hat e_{r'}$ is the unit retarded position vector.
Now since the first two terms fall as $\dfrac{1}{r^{'2}}$ and the third term $\dfrac{1}{c^2}\dfrac{d^2\hat e_{r'}}{dt^2}$ by $\dfrac{1}{r'}$, only the third term is significant at large distances from the charge, and I now quote:
This term permits a charged particle to influence another charged particle at a great distance through the 1/r electric field. This is referred to as electromagnetic radiation and light is a familiar example of this phenomenon. It is obvious from the formula that only accelerating charges produce radiation.
Now, my questions:
- Why is the "at large distances" constraint needed? If the distance is small then the first two terms will also influence the radiation right?
- If the charge is in uniform motion, then the part of the third term $\dfrac{1}{c^2}\dfrac{d^2\hat e_{r'}}{dt^2}$ which falls as $\dfrac{1}{r'}$ is 0 but the first two terms are not. So, at large distances, even though very small, can non-accelerated charges produce radiation?
EDIT: Here's the link to the textbook: Physics I: Oscillations and Waves notes by S. Bharadwaj and S.P. Khastgir, https://drive.google.com/file/d/1CTGX7fXzeb_rHn5oqDto-3REWJKoR83S/view?usp=sharing. I am referencing pg 61.