If earth's magnetic field disappeared, would cosmic radiation lead to increased radioactivity? This is about the effect of cosmic radiation on earth. Is it the type of radiation that could make things radioactive?
So if earth's magnetic field weakened considerably (such as could happen if it was continually flipping rapidly), or disappeared altogether, would it cause an increase in radioactivity? Could it cause an increase in, for example, Carbon-14?
 A: Yes, it will cause many things to become radioactive, carbon-14 being one example. The cosmic radiation itself may reach the ground so we have much more to worry than just carbon-14. One of the first large-scale phenomena you will see is global aurora caused by solar wind. Over time, solar wind will disrupt the ozone layer and destroy yet another shield protecting us from cosmic and solar radiation. Ultimately, Earth's atmosphere will be stripped away by solar wind and Earth will become barren like Mars.

Update:
I just realized I should emphasize that my answer is based on complete elimination of the magnetosphere.
A: The flux of cosmic particles that is particles that come from outside our solar system  is linked to the suns magnetic field; the suns field to a degree protects the earth from cosmic particles  more so than the earths field.
Our stars magnetic field has a large influence on shielding the earth from cosmic rays but the suns field is not a constant, it also varies like the earths does.
Additionally the earth magnetic field is influenced by changes in the suns magnetic field.
Thing would not become more radioactive, and the flux of inbound solar radiation would not change.  What would increase is the penetration depths and distribution pattern of the energetic particles coming from the sun.
Inbound High energy particles are responsible for the formation of complex molecules, so perhaps there would be an increase in C-14; but no to 'radioactivity' directly, indirectly yes.  
