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What's the equation of kinetic energy of charged particle?

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I was looking for equation of motion. I came up with a solution but it doesn't satisfy me. Cause I was trying to find motion of that particle using Lagrangian. We know that $$W=\int \vec F\cdot d\vec l$$

$W=T$ for some cases and I came up with $T=qV$. In Euler-Lagrange, kinetic energy has velocity as function, in $T=qV$ there's no velocity directly, the equation actually tells me that particle is gaining kinetic energy from potential (more precisely, potential is converting into kinetic). At first sight, I wrote that $T=\frac{1}{2}m\ddot{r}^2$ what if particle is massless(?) so it's not very helpful. Where I took $$L=0.5m\ddot{r}^2-\frac{1}{4\pi\epsilon_0}{q}{r}$$ if I try solve Euler-Lagrange using that Lagrangian then I get $m\ddot{r}=\vec E$. How force is equal to electric field? It totally doesn’t make any sense to me, their dimension doesn't match either. None of these equation satisfy me. what I think that is I got wrong result for taking kinetic energy which doesn’t apply to charged particle. So what's the kinetic energy of charged particle?

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2 comment threads

Define your variables. (2 comments)
What's the kinetic energy of a massless particle? (3 comments)

Comments on What's the equation of kinetic energy of charged particle?

What's the kinetic energy of a massless particle?
Canina‭ wrote 10 months ago:

Not a physicist, so I might be completely wrong here, but for your rebuttal of $T=\frac{1}{2}m\ddot{r}^2$, consider: What is the kinetic energy of a massless particle? Does a massless particle have any kinetic energy?

deleted user wrote 10 months ago:

I believe they have. Let's take photon(s), they are massless but moves too fast. And they have kinetic energy for their motion(?).

maybe I can plug in relativistic mass for massless particle (according to some derivation massless particles can have relativistic mass) but I am not going to use that term until I get clear idea of it

celtschk‭ wrote 10 months ago:

deleted userk The formula for kinetic energy you used is only for non-relativistic particles. It definitely cannot be applied to photons. Photons do have kinetic energy (indeed, all of their energy is kinetic energy), but it isn't given by that formula.

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