Final answer:
The time spent by the proton inside the magnetic field is directly proportional to the velocity of the proton (option D).
The radius R of a proton's helical path in a magnetic field is determined by the perpendicular component of its velocity and magnetic field strength, while the pitch p is calculated based on the parallel component of velocity and the period of the circular motion.
Step-by-step explanation:
When a proton enters a region of a uniform magnetic field at an angle to the magnetic field, it experiences a force that causes it to move in a helical path.
The radius of the helix, R, depends on the component of velocity that is perpendicular to the magnetic field, while the pitch, p, (the distance between turns), is determined by the component of velocity parallel to the field.
Finding the Radius R
The magnetic force acting on the proton provides the centripetal force required to make the proton move in a circular path. Using the relationship Fc = qvℓB, where Fc is the centripetal force, q is the charge of the proton, vℓ is the component of velocity perpendicular to B, and B is the magnetic field strength, we can solve for the radius R using R = mvℓ/(qB).
Finding the Pitch p
The pitch p of the helix can be found by considering the velocity component parallel to the magnetic field, which is not affected by it.
Given the period T of the circular motion, which is the time to make a full circle, p = v//T.
The time T can be obtained by dividing the circumference of the base circle of the helix (2πR) by the perpendicular component of velocity, T = 2πR/vℓ.