Force Laws and Maxwell's Equations
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Returning again to equation (1a), we see that in the absence of a gravitational field the force on a particle with q = m = 1 and velocity v at a point in space where the electric and magnetic field vectors are E and B is given by
##\ \ \ \ \ \mathbf f = \mathbf E + \mathbf v \times \mathbf B ##
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Thus if the particle is stationary with respect to the original x,y,z,t coordinates, the force on the particle has the components
## \ \ \ \ \ f_x = E_x \ \ \ \ \ f_y = E_y \ \ \ \ \ f_z = E_z ##
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Therefore, from equation (9), we see that the transformed components of the total electromagnetic force are
##\ \ \ \ \ f_{x'} = f_x \ \ \ \ \ f_{y'} = \sqrt{1-v^2} f_y \ \ \ \ \ f_{z'} = \sqrt{1-v^2} f_z \ \ \ \ \ \ ## (10)
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Just as the Lorentz transformation for space and time intervals shows that those intervals are the components of a unified space-time interval, these transformation equations show that the electric and magnetic fields are components of a unified electro-magnetic field. The decomposition of the electromagnetic field into electric and magnetic components depends on the frame of reference.
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we see that Maxwell's equations are invariant under Lorentz transformations. Moreover, any physical force consistent with special relativity must transform in accord with (10), because otherwise a comparison of the forces in different frames of reference would give different results.