Energy density (electrodynamics/ Dirac delta etc)

[sunrah]

So I have the following velocity vector of a charged particle in an EM field

$\dot{\vec{r}} = (v_{0x}cos(\alpha t) - v_{0z}sin(\alpha t), \frac{qEt}{m} + v_{0y}, v_{0z}cos(\alpha t) + v_{0x}sin(\alpha t))$


and I have to state the energy density, which is defined as follows:


$\tau = \sum \frac{m_{i}}{2}\dot{\vec{r_{i}}}^{2}\delta(\vec{r}-\vec{r}_{i})$

My question is whether all I have to do is substitute the vector in the definition of the energy density or if I have to do something with the Dirac distribution as well. Thanks

 

[anathema]

your understanding of the energy density equation is correct. You can simply substitute the given velocity vector into the definition of energy density. The Dirac distribution does not need to be taken into account in this case, as it is used to represent a point charge and is not necessary for calculating the energy density of a charged particle in an EM field. However, if you are dealing with a system of multiple charged particles, then you may need to consider the Dirac distribution in order to accurately calculate the energy density.