Chapter 27, Problem 27.57P

A particle with negative charge q and mass m = 2.58 × 10⁻¹⁵ kg is traveling through a region containing a uniform magnetic field $\stackrel{\to }{\text{B}}$ = −(0.120T) $\hat{k}$. At a particular instant of time the velocity of the particle is $\stackrel{\to }{\upsilon }$ = (1.05 × 10⁶m/s)(−3î +4ĵ + 12 $\hat{k}$) and the force $\stackrel{\to }{\text{F}}$ on the particle has a magnitude of 2.45 N. (a) Determine the charge q. (b) Determine the acceleration $\stackrel{\to }{\text{a}}$ of the particle, (c) Explain why the path of the particle is a helix, and determine the radius of curvature R of the circular component of the helical path, (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the x- and y-coordinates do vary in a periodic way. If the coordinates of the particle at t = 0 are (x, y, z) = (R, 0, 0), determine its coordinates at a time t = 2T, where T is the period of the motion in the xy-plane.