When I was reading the first volume of the theoretical minimum by Leonard Susskind, I encountered upon this:
But when I checked Griffiths and all my other electricity and magnetism books, it showed $$\mathbf F=q({\mathbf v} \times \mathbf B)$$ instead. Where did the extra factor of $c$ come from? Which one is the correct equation? Are both equations equivalent? Or was that a misprint in Susskind's book?
It depends on the units of electromagnetism you are using. The straight-$\vec v$ form is characteristic of SI-style units where $\nabla\times E=-\dot B$ while the cgs-style units where $\nabla\times E=-\dot B/c$ have $\vec v/c.$
I actually don't know a great way to derive this from the Maxwell's equations directly, but if you know some relativity then you can get it indirectly by looking at $$F^{\mu\nu}=\partial^\mu A^\nu -\partial^\nu A^\mu.$$
