The answer is in two parts:
- The first parts is in response to the original question (before it
was edited), which was: "Is the output of an induction coil
AC or DC?"
- The second part is in response to the second form of the question as
of 2018-02-07 (7th Feb): "Why does a Crookes tube work when connected to an induction coil?".
Part 1, in answer to "Is the output of an induction coil AC or DC?":
Your question is relevant in the face of evidence. Firstly: what is the definition of alternating current? Quoting Wikipedia:
"Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction."
In response to the primary coil input current i$_1$, the secondary coil has an output voltage $V_2$, with a waveform like that in the figure below (with and without a capacitor across the interrupter). Credit to Wikipedia.
By the definition above, this is clearly this is alternating current, though not having a clean wave form like the AC we normally use in our homes.
Is this a peculiarity of an induction coil, that it outputs AC instead of DC? Not really: to induce voltage in a secondary coil, the voltage in the primary coil (more exactly the magnetic field it causes) must be changing, or the secondary coil must be in motion (as in a generator). This is electromagnetic induction.
To get direct current from an induction coil or a voltage transformer (same principle), we normally employ a rectifier such as a Wheatstone bridge to effect the change as in the figure below, where the AC on the left would come out as AC on the right, were it not for the array of rectifiers shown. The rectifiers themselves do not create DC but a pulsating voltage as seen on the right. This needs to be smoothed using appropriate capacitors to become DC (illustration credit to Wikipedia):
Part 2, in answer to "Why does a Crookes tube work when connected to an induction coil?":
If a voltage is alternating (AC), but during a particular period it remains long enough with one polarity or direction (+ or -), such that the circuit to which it is applied cannot differentiate it from 'stable DC' during that period, then, as far as the circuit is concerned, the voltage is 'DC' during that period.
In the case of the Crooke's tube, the voltage induced by an induction coil attached to it needs to satisfy the following conditions:
- It must supply a high enough voltage to the Crooke's tube, charging the anode and cathode with high '+' and '-' polarities respectively.
- This high voltage must be maintained long enough to cause ionisation of the gas inside the Crooke's tube and produce an electrical discharge in the form of a stream of electrons (a cathode ray).
It turns out that the charge on the anode and cathode in the Crooke's tube can be reversed, and the glow of electrons in it will still be visible. Therefore a Crooke's tube will work using symmetric AC, provided again that during each half cycle, the two conditions mentioned above are met.