So if I stick a cathode of at one end of a plasma column and an anode at the other and turn on the power, will the cathode necessarily explode?
It most certainly will, if you don't allow the charges to go somewhere else.
If the Sun were as paravolt has described, there would be nowhere for the charges to go (the current flow, in his simple idea, is from heliosphere - actually heliopause or heliosheath most likely - to Sun, probably photosphere, but he didn't specify).
For the charges to not accumulate, you need a magic electric element never created in any lab on Earth, a
Mozode:
What, then, is the Mozode?
First and foremost, it is a critically important element in an electric circuit; its electrical nature is essential.
Second, it is spherical, or approximately spherical.
Third, its surface is composed of Moplasma.
Fourth, it emits electrons and protons in equal quantities*.
Fifth, it has a voltage of ~600,000,000 volts, with respect to a hollow sphere which completely encloses it (the Mozode is at the approximate centre of this sphere; this outer sphere may be another Moz-thingie).
Sixth, it has not been simulated in real science experiments here on Earth; it does not meet the burden of proof from the standpoint of empirical physics in any lab on Earth; (yet get the idea).
*
actually, electrons and positive ions (the ions are predominantly protons and helium ions)