A particle of charge q and mass m is moving in a region where the electric field

Question

A particle of charge q and mass m is moving in a region where the electric field, E- ExEy magnetic field, B - Bz, are uniform. Show that its position can be written as, E?, and the where vo is a constant velocity, which satisfies B x Vo-Ek+Eyy. R is an arbitrary radius, ? s an arbitrary (constant) angle and ae-qB/m is the cyclotron frequency. Show that the x-y component of vo is E x ?/B and its z component is arbitrary. The arbitrary constants, R, ?, the z component of vo and the vector ro, are determined by the initial values of the particle velocity and position.

Explanation / Answer

given charge of particle = q
mass of particel = m
electric field = E
E = Ex i + Ey j + Ez k
magnetic field B
B = B k
both magnetic and electirc fields are uniform

so, let initial position of particle be ro
intiial velocity be vo
then
r(t) = ro + vo*t + 0.5*a*t^2
here a is acceleration of the particel
now, force on particle = F
F = q v x B + qE
hence
a = q(v x B + E)/m
now, v is constnat velocity, vo
and -vo x B = Ex i + Ey j
hence
a = q(-Exi - Eyj + Exi + Eyj + Ezk )/m
a = q(Ez/m)k
hence

r(t) = ro + vo*t + (q/2m)Ez k *t^2

now inside the cyclotron, the particle is also moving in circular orbit at radius R where R is radius of cyclotron
hence
the coordinates of particel in cyclotron = R(cos(wc*t + phi)i + sin(wc*t + phi)j)
where wc is cyclotron frequency, wc = qB/m
and hence
fonal locaiton of aparticle is
r(t) = ro + vo*t + (q/2m)Ez k *t^2 + R(cos(wc*t + phi)i + sin(wc*t + phi)j)
where i and j are unit vectors along x, y axis

as r(t) does not have z component, it is not relevent in the kinematics of the problem and hence it is arbitrary

here , R is initial radius of the partile in cyclortron, and phi is the intiial angle it made with the axes inside the cyclotron

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