It is experimentally known that the equation of motion for a charge e moving in a static electric field E is given by: d/dt (gamma m v)=eE

ntaraxq 2022-07-13 Answered
It is experimentally known that the equation of motion for a charge e moving in a static electric field E is given by:
d d t ( γ m v ) = e E
Is it possible to show this using just Newton's laws of motion for the proper frame of e, symmetry arguments, the Lorentz transformations and other additional principles?
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Answers (2)

lydalaszq
Answered 2022-07-14 Author has 11 answers
If for some reason you'd like an explicitly relativistic formulation, take a look at the Lorentz force law:
d p μ d τ = e v ν F μ ν
For the derivative with respect to coordinate time that you want, we need to multiply through by d τ / d t = 1 / γ. But for a constant electric field in Cartesian coordinates, the only nonzero components of F μ ν are F 0 a = F a 0 for a = 1 , 2 , 3 , which are the electric field components. Thus, only the v 0 = γ term can contribute, canceling factor brought in by time dilation factor. QED.
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Cierra Castillo
Answered 2022-07-15 Author has 6 answers
I think this is a lot simpler than you suspect. It's really just Newton's 2nd law, and recognising the concept of momentum.
d d t ( γ m v ) = d p d t = F = e E
Since classically the electric field E is defined as the force F divided by the elementary charge e.
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