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mysearch
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I am trying to establish whether the force defined by the Lorentz equation below is invariant under the Lorentz transforms:
[1] [tex]F = F_E + F_B = qE + qvB [/tex]
In the context of this equation, [q] is moving with velocity [v] such that it is acted on by both an electric E-force and magnetic B-force. If this frame of reference were transformed into a stationary (*) frame such that v=0, then [tex]F_B*[/tex] would be zero, i.e.
[2] [tex]F* = (F_E*) + 0 [/tex]
So is the Lorentz force [F] invariant?
If so, I presume you can equate [1] and [2] as follows
[3] [tex]F = F_E + F_B = (F_E*) + 0 = F* [/tex]
Which would lead to:
[4] [tex]F_E* = F_E + F_B [/tex]
However, I believe the magnetic force can be related to the electric force as follows:
[5] [tex]F_B=F_E \frac {v^2}{c^2}[/tex]
Again, if this is correct, it would appear that [4] can be transposed as follows:
[6] [tex]F_E* = F_E (1+ \frac {v^2}{c^2}) [/tex]
As such, the implication appears to be that [tex]F_E* > F_E [/tex] and the invariance of the Lorentz force [F=F*] leads to the conclusion that the electrostatic force [tex]F_E[/tex] in the stationary frame is transformed into [tex]F_E+F_B[/tex] in the moving frame. Would welcome any clarification of these points. Thanks
[1] [tex]F = F_E + F_B = qE + qvB [/tex]
In the context of this equation, [q] is moving with velocity [v] such that it is acted on by both an electric E-force and magnetic B-force. If this frame of reference were transformed into a stationary (*) frame such that v=0, then [tex]F_B*[/tex] would be zero, i.e.
[2] [tex]F* = (F_E*) + 0 [/tex]
So is the Lorentz force [F] invariant?
If so, I presume you can equate [1] and [2] as follows
[3] [tex]F = F_E + F_B = (F_E*) + 0 = F* [/tex]
Which would lead to:
[4] [tex]F_E* = F_E + F_B [/tex]
However, I believe the magnetic force can be related to the electric force as follows:
[5] [tex]F_B=F_E \frac {v^2}{c^2}[/tex]
Again, if this is correct, it would appear that [4] can be transposed as follows:
[6] [tex]F_E* = F_E (1+ \frac {v^2}{c^2}) [/tex]
As such, the implication appears to be that [tex]F_E* > F_E [/tex] and the invariance of the Lorentz force [F=F*] leads to the conclusion that the electrostatic force [tex]F_E[/tex] in the stationary frame is transformed into [tex]F_E+F_B[/tex] in the moving frame. Would welcome any clarification of these points. Thanks