Q&A

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

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I was thinking to find kinetic energy for a charged particle. I came up with a solution but it doesn't satisfy me. Cause I was 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 $T=\frac{1}{2}m\ddot{r}^2$ I told me what if particle is massless(?) so it's not very helpful and if I look solve Euler-Lagrange using that kinetic energy then I get $m\ddot{r}=\vec E$ it's totally wrong, their dimension doesn't match. None of these equation satisfy me. So what's the equation of kinetic energy of a charged particle?

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