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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 $$W=\int \vec F\cdot d\...
#2: Post edited
I was thinking to find kinetic energy for a charged particle. I came up with a solution but it doesn't satisfy me. Cause I was to find motion of that particle using Lagrangian. We know that- $$W=\int \vec F\cdot d\vec l$$
$W=T$ for some cases and I came up with $T=qV$. In Euler-Lagrange, kinetic energy has velocity as function, in $T=qV$ there's no velocity directly, the equation actually tells me that particle is gaining kinetic energy from potential (more precisely, potential is converting into kinetic). At first sight, I wrote that $T=\frac{1}{2}m\ddot{r}^2$ I told me what if particle is massless(?) so it's not very helpful and if I look solve Euler-Lagrange using that kinetic energy then I get $m\ddot{r}=\vec E$ it's totally wrong, their dimension doesn't match. None of these equation satisfy me. So what's the equation of kinetic energy of a charged particle?
- I was looking for equation of motion. I came up with a solution but it doesn't satisfy me. Cause I was trying to find motion of that particle using Lagrangian. We know that
- $$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}$$ if I try solve Euler-Lagrange using that Lagrangian then I get $m\ddot{r}=\vec E$. How force is equal to electric field? It totally doesn’t make any sense to me, their dimension doesn't match either. None of these equation satisfy me. what I think that is I got wrong result for taking kinetic energy which doesn’t apply to charged particle. So what's the kinetic energy of charged particle?
#1: Initial revision
What's the equation of kinetic energy of charged particle?
I was thinking to find kinetic energy for a charged particle. I came up with a solution but it doesn't satisfy me. Cause I was to find motion of that particle using Lagrangian. We know that
$$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?