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1) In F = qVxB, V is the velocity of the particle whose charge is q; B is
the magnetic field (of some source) measured at the location of q; F is the
force which q experiences due to its interaction with the field B. All of
these four quantities are measured by an observer in a given inertial frame.
In particular, V is simply the velocity of the charged particle in this
frame. (Note that the kinematical status of the source of B is not
explicitly relevant - the local [at q] value of its field B says everything
about its magnetic force on q - this is the essential usefulness of a field
theory - indeed B may be the net field of numerous sources.)