Final answer:
The radius of the proton's helical path is found using the formula R = mv/(qB), considering only the velocity component perpendicular to the magnetic field, which is v_x in this case. We use the known mass and charge of a proton along with the given velocity and magnetic field strength to calculate R.
Step-by-step explanation:
To find the radius of the proton's helical path when it enters a magnetic field that is directed in the y-direction, with initial velocity components vx = 1.00 × 10^4 m/s and vy = 7.50 × 103 m/s, we need to consider the motion in the plane perpendicular to the magnetic field. The component of velocity that is perpendicular to the magnetic field is vx, and this will determine the circle part of the helical path. The magnetic force acting on the proton provides the centripetal force needed for this circular motion.
The magnetic force F on a charged particle moving in a magnetic field is given by F = qvBsin(θ), where q is the charge of the proton, v is the velocity, B is the magnetic field strength, and θ is the angle between the velocity and magnetic field. Since the proton's velocity in the y-direction will not cause additional force in the perpendicular direction of the motion, we can focus only on the x component for finding the radius.
The formula to find the radius R of the circular path is R = rac{mv}{qB}, where m is the mass of the proton, v is the velocity perpendicular to the magnetic field (vx), q is the charge of the proton, and B is the strength of the magnetic field. Using the mass (m) of a proton as 1.67 × 10^-27 kg, charge (q) as 1.60 × 10^-19 C, and given vx and B values, we calculate the radius R of the helical path.