Several charges enter a uniform magnetic field directed into the page. (a) What path would a positive charge q moving with a velocity of magnitude v follow through the field? (b) What path would a positive charge q moving with a velocity of magnitude 2v follow through the field? (c) What path would a negative charge -q moving with a velocity of magnitude v follow through the field? (d) What path would a neutral particle follow through the field

We know that in a uniform field a particle with initial velocity that is perpendicular to the field moves in a circle and having a radius R. Which depends on the magnetic field B and the particle mass m. And have speed v and charge q. This can be written as the formula:

$R=\frac{mv}{\left|q\right|B}$

However, if and when the particle has a velocity in arbitrary direction then it will be obtained a helicoidal motion. Now according to the velocity component in the direction of the magnetic field. Now we can imagine that the helicoidal motion is superposition of circular motion due to velocity components.

From the above definition we can conclude that the motion of the positive charged particle in a uniform magnetic field, will be a helicoidal and having a circular direction counter clock wise. Since the charge is positive the radius of one circle incorporated in helicoid is given by:$R=\frac{mv}{\left|q\right|B}$

Now the motion of the negative charged particle in the uniform magnetic field, will be a helicoidal with circular direction and clock wise, the radius of the circle will be same the above one. The motion will be same for the velocity perpendicular to the magnetic field. But instead, counter clock wise the motion will be clock-wise.

No, there is no magnetic force act on a neutral particle, therefore the particle will continue with the motion.