Friday, 6 May 2016

electromagnetism - An example which contradict to Newton's 3rd law?


Let a,b be two charged particles. ra(0)=0

rb(0)=rˆj
va(t)=vaˆi
vb(t)=vbˆj


In which both va and vb $<

Then


Eab(0)=qa4πϵr2ˆj


Bab(0)=μqava4πr2ˆk


Eba(0)=qb4πϵr2ˆj


Bba(0)=0


Note that va and vb $<

Hence


Fab(0)=qb(Eab+vb×Bab)

=qaqb4πϵr2ˆjμqavavb4πr2bˆi


But


Fba(0)=qaqb4πϵr2ˆj



Consequently FabFba


This result contradict to Newton's 3rd law!! But I cannot find any error... It troubled me.



Answer



The details of your analysis are not quite right - that's not what the electric field of a moving charge looks like, for example. This is probably because you haven't learned all the rules of electromagnetism yet. Still, the spirit of your question is hitting at an important point.


Charges do not conserve momentum and don't obey Newton's third law. You have to include the momentum of the electromagnetic field to see conservation laws hold.


There's an accessible discussion in section 8.2 of Griffiths "Introduction to Electrodynamics" if you would like a little more math.


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