Please help with part C The answer is NOT 123 Thanks The fig

Please help with part C! The answer is NOT 1=2=3 Thanks!!

The figure below shows the paths flowed by three charged particles through a region of uniform magnetic field that is directed perpendicular to the page. The particles have the same mass, and their charges have the same magnitude. The paths of particles 1 and 2 represent one quarter of a circle, while that followed by particle 3 is a half circle. (a) Rank these particles, from largest to smallest, based on their speed. (Use only \">\"or\"symbols. Do not include any parentheses around the letters or symbols.) 2-1>3 (b) Rank these particles, from largest to smallest, based on the magnitude of the magnetic force they experience in this region. (Use only\">\" or\"symbols. Do not include any parentheses around the letters or symbols.) 2 1>3 (c) Rank the particles based on the time they spend passing through the region shown in the picture, from argest to smallest. (Use only \">\"orsymbols. Do not include any parentheses around the letters or symbols.)

Solution

given, region of uniform magnetic field
particles have same mass m, same charge magnitude q

now, for a charged particle of mass m moving with speed v
qvB = mv^2/R [ where R is radius of the circle followed by the particle in the uniform magnetic field B]
hence
mv/qB = R

a. ranking from largest to smallest based on speed
   v = RqB/m
   m is same for all particle3s and so is q and B
   so v is proiportional to R
   from the diagram
   R2 > R1 > R3
   hence
   2 > 1 > 3

b. based on magnertic force from largest to smallest
   magnetic force = qvB
   as q and B is same for all particles
   2 > 1 > 3

c. time spent passing throught thje region = distance/speed
   so from diagram
   R1 = 2
   R2 = 3
   R3 = 1

   so, v1 = 2k [ where k = qB/m]
   v2 = 3k
   v3 = k

   so t1 = 2*pi*R1/4*v1 = pi/2k
   t2 = 2*pi*R2/4v2 = pi/2k
   t3 = 2*pi*R3/2v3 = 2*pi*1/2*k = pi/k
   hence
   t3 > t2 = t1
   3 > 2 = 1