At the world's largest synchrotron, the Large Hadron Collider $($LHC$),$ lead $($ Pb $)$ nuclei $(m_{Pb}=3\mathrm{.}44{10}^{-25}kg)$ are accelerated along circular trajectories measuring $26\mathrm{.}7$ km in circumference using magnetic fields to guide them along their path. During tests runs, these nuclei are sent around the ring at a rate of $563$ revolutions per second. Assume a lead nucleus has a charge equivalent to $82$ protons. $(6)$

a$.$ Determine the magnitude of the magnetic force that must be exerted on a Pb nucleus in order to achieve that rate of motion and draw the force vector on the umbra.... $(<mi>)</mi>$

b$.$ Calculate the magnitude of the magnetic field necessary to guide a Pb nucleus along the $26\mathrm{.}7$ km circumference, draw its vector in the diagram and label it $B_{Pb}.(2)$

c$.$ If$,$ instead of a Pb nucleus, electrons were sent through the LHC in the same direction an the same rate of $563re\frac{v}{s},$ what magnitude of field would be necessary? Draw the new field vector $B_{e_{-}}$in the diagram. $(2)$