The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation. If the electric field vector is E and the magnetic field is B, the force on a particle of mass m that carries a charge q and has a velocity v F = qĒ + qỡ x B where we assume v << c (speed of light). (a) If there is no electric field and if the particle enters the magnetic field in a direction perpendicular to the lines of magnetic flux, show that the trajectory is a circle with radius mv qB where we = qB/m is the cyclotron frequency. (b) Choose the z-axis to lie in the direction of B and let the plane containing E and B be the yz-plane. Thus v r = where 9 Wc B = BÊ, E = Eyĵ + E₂k. Show that the z component of the motion is given by z(t) = zo + żot + qEz 2m t. z(0) = zo and ż(0) = żo.
The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation. If the electric field vector is E and the magnetic field is B, the force on a particle of mass m that carries a charge q and has a velocity v F = qĒ + qỡ x B where we assume v << c (speed of light). (a) If there is no electric field and if the particle enters the magnetic field in a direction perpendicular to the lines of magnetic flux, show that the trajectory is a circle with radius mv qB where we = qB/m is the cyclotron frequency. (b) Choose the z-axis to lie in the direction of B and let the plane containing E and B be the yz-plane. Thus v r = where 9 Wc B = BÊ, E = Eyĵ + E₂k. Show that the z component of the motion is given by z(t) = zo + żot + qEz 2m t. z(0) = zo and ż(0) = żo.
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Transcribed Image Text:The motion of a charged particle in an electromagnetic field can be obtained from the
Lorentz equation. If the electric field vector is E and the magnetic field is B, the force
on a particle of mass m that carries a charge q and has a velocity v
F = qĒ + qỡ x B
where we assume v << c (speed of light).
(a) If there is no electric field and if the particle enters the magnetic field in a direction
perpendicular to the lines of magnetic flux, show that the trajectory is a circle with
radius
mv
qB
where we = qB/m is the cyclotron frequency.
(b) Choose the z-axis to lie in the direction of B and let the plane containing E and B be
the yz-plane. Thus
v
r =
where
9
Wc
B = BÊ, E = Eyĵ + E₂k.
Show that the z component of the motion is given by
z(t) = zo + żot +
qEz
2m
t.
z(0) = zo and ż(0) = żo.
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