Description of the motion of the COM of a system of charged particles moving in a magnetic field 
I recently came across a problem where an electrically neutral particle is at rest in a uniform magnetic field. The particle now splits into two charged particles of equal mass $m$; charge and mass being conserved. Now, we know that if one particle has charge $q$, the other must have equal and opposite charge $-q$. It is said that the two particles collide after some time and we are asked to find that time. 

My reasoning is that since the center of mass(com) of the system remains stationary, the particles will split in opposite directions to each other and thus collide just when each particle completes a semi-circle due to the magnetic force. With this approach, I get the answer right. Now let's forget the time period part and analyze the motion of the particles.
Second case: In analyzing the motion of the particles from a different perspective, we can say that throughout the motion, the center of mass must remain at rest as its charge is zero (q+(-q)=0) and so magnetic force is zero. So the charged particles must continue to move opposite to each other just so that the com remains stationary and not in a semi-circle. Perhaps the forces exerted on each other do the job. I am actually confused about the properties and motion of the com in situations involving charged particles in magnetic fields.  I wonder whether there is a center-of-charge kind of concept where that point remains stationary when a neutral particle splits into two or more charged particles ? . Such a concept will simplify problem solving to a great extent in the same way the concept of com does.
Can someone please clarify these to me:


*

*Why is the second case incorrect ?

*Is there such a center-of-charge concept ?


Please feel free to edit and add relevant tags to the question.  
 A: 
Is there a center-of-charge kind of concept where that point remains stationary when a neutral particle splits into two or more charged particles ?

Not when there are external forces acting on the system. As soon as the particle breaks down, the two charged particles experience magnetic force in the same direction, so their sum momentum changes and the center of mass moves as well.
A: Yes,the particles will have attractive forces between each other ,which can and will disturb their trajectories . But as far as the base question is concerned, I think we are supposed to neglect that interaction (as I saw another version of this problem which said so) .
   I think your first view point is better .Basic reason being that the initial momentum(net) of the system is zero, and momentum (net) should be conserved so after breaking they move antiparallel to each other. And as they move in the B-field they are both get curved in a good circle and collide (circle only after neglecting the Electro-forces)
