Does the force of magnetism cause electricity?
The magnetic field and the electric field are both part of a single electromagnetic field, governed by Maxwell's equations.
Maxwell's equations tell us the fundamental relation between magnetism and electricity.
eg a moving magnet will "induce" an electric current in a conductor.
Yes, if it is changing. It's called Faraday's law of induction. This is the basic operating principle behind electric generators. As tiny-tim mentioned, the electric field and magnetic field are really different faces of the same thing. If you move a wire near a magnet, the magnetic field exerts a force directly on the moving charges. If you move a magnet near a wire, the moving magnetic field creates (induces) a curling electric field, and the electric field exerts a force on the stationary charges in the wire. It turns out that these two situations are identical, they are just seen from different frames. If you ask, "Which field is really creating the current, the magnetic field or the induced electric field?" then the answer is "both". They are both part of the same thing, and it only depends from which frame you are looking which one you see more of.
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