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Phyrrus
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Homework Statement
Assume the Earth's magnetic field is almost homogeneous with direction along the z-axis, with a small inhomogeneous modification which make the field lines converge towards the z-axis. Also ignore relativistic and gravitational effects.
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First assume the magnetic field, B = B[itex]_{0}[/itex] = B[itex]_{0}[/itex]k, to be time independent and homogeneous, with
k as a unit vector in the z-direction. A particle with charge q and mass m is moving in this field.
Initially, at time t = 0 the particle has velocity v[itex]_{0}[/itex], with u[itex]_{0}[/itex] as the component in the z-direction and
w0 as the component in the x; y-plane.
a) Write the vector form of the equation of motion of the particle and show that it has solutions of
the form
r(t) = ρ[itex]_{0}[/itex](cos [itex]\omega[/itex][itex]_{0}[/itex]ti + sin [itex]\omega[/itex][itex]_{0}[/itex]0tj) + v[itex]_{z}[/itex]t k
Determine the constants ρ[itex]_{0}[/itex], ω[itex]_{0}[/itex] and v[itex]_{0}[/itex] in terms of the initial velocity and magnetic field strength B0.
Homework Equations
F=q(E+v[itex]\times[/itex]B)
The Attempt at a Solution
E=0
a=(q/m)(v[itex]_{0}[/itex][itex]\times[/itex]B[itex]_{0}[/itex])
v[itex]_{0}[/itex]=(w[itex]_{0}[/itex],u[itex]_{0}[/itex])
where w[itex]_{0}[/itex]=|w[itex]_{0}[/itex]|cos[itex]\vartheta[/itex]i+w[itex]_{0}[/itex]sin[itex]\vartheta[/itex]j)
therefore v[itex]_{0}[/itex][itex]\times[/itex]B[itex]_{0}[/itex] = B[itex]_{0}[/itex]|w[itex]_{0}[/itex]|sin[itex]\vartheta[/itex]i-B[itex]_{0}[/itex]w[itex]_{0}[/itex]cos[itex]\vartheta[/itex]j