Single Particle Dynamics in EM Fields.

[peterjaybee]

Hello, I am having trouble seeing how to derive the energy equation for a charged particle from the equation of motion.

The equation of motion is

$$m\frac{d\bar{v}}{dt}=q(\bar{E}+\bar{v} \times \bar{B})$$

Then in the notes I have it says "projecting the eq. of motion onto the particles velocity vector leads to the particles energy equation:"

$$\frac{d}{dt}\left(\frac{1}{2}m\bar{v}^{2}\right)=q\bar{E}\cdot\bar{v}$$

Could someone please take me through the steps inbetween these two equations, or explain what is meant by projecting the equation of motion onto the particles motion please.

Many Thanks,

Peter

 

[Meir Achuz]

Dot the first equation with the vector v, and you will get the second equation after performing the time differentiation of 1/2 mv^2.

 

[peterjaybee]

Got it, Thanks