Can electricity transfer radioactivity? If a cable used to power something is exposed to a radioactive source will it over time make the entire cable radioactive? 
 A: Electricity is the movement of electrons. Electrons cannot become radioactive. So a current in the cable will not change the radioactivity of the cable.
A: A cable cannot transport radioactivity away from the reactor using electrical current as a transport mechanism. 
Nevertheless - if a material is exposed to a source of particles of sufficiently high energy (high energy particle beam, or neutron beam, for instance), the material can become activated (meaning that some of the atoms in the material will become radioactive), and may remain activated for a very long time, so use of materials that had been previously used in particle beam applications must be carefully considered.
So, the answer is yes, in the sense that material can become radioactive from exposure to high energy particles as one would find in a reactor/particle beams and that activation can be long lived and could potentially migrate through reuse/re-purposing of components, but no in the sense that the continued activation is dependent on the local presence of such a high energy source. 
A: Are you worried that the cables that go to the Fukushima reactors will carry radioactivity out?
The answer is No. You should read up a bit on radioactivity and educate yourself, since it is one of the facts of life. In the article you will see that it is atoms that are responsible for radioactivity whereas the current in the cables is due to electrons. The parts of the cables that are in a radiation environment will become radioactive, but that activity will remain in the locality, and the length of cable that was exposed. It in no way can be transmitted away from the region the way the current is transmitted.
