First you need to know that the three phases are not all in-phase. There is a time lag/phase shift between the peak of Brown and Black, and similarly between Black and Grey. It is because in a three phase alternator, the coil to generate the Brown voltage is say at the 12 o'clock and that for Black and Grey are at 4 0'clock and 8 o'clock. So when they rotate together on the rotor axis there is a constant phase shift albeit they all generate the same sinusoidal waveform.
In terms of angular degrees, the 12 0'clock could be called 0 degrees, then the other two moving around clockwise would be 120 and 240 degrees.
These phase voltages drive the currents that flow. If the loads on each phase are the same then the currents will be the same magnitude but again time/phase shifted by 4 hours/120 degrees.
Here is the exercise for you:
If you take any position of the axis at a particular time (say 90 or 3 0'clock) degrees for Brown) and work out sin 90 and add it too sin(90 + 120) and sin (90 + 240) - the other two phases you will find that they answer is zero. Try replacing 90 by any angle x between 0 and 360 degrees - whatever angle x you pick, if the load is balanced all the phase currents will sum to zero. So for a balanced load the neutral does not carry any current - for an unbalanced load it carries the residual current which is that current which is
left over after the parts of the phase currents which do add up to zero.
It is easier to see this using phasors/vectors. Take a look at:
I have only quickly looked at this youtube clip but there are others if you google 'three phase neutral currents' or something similar.
