Particle moving perpendicular to a constant magnetic field
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A particle of mass m and charge q is moving with speed v perpendicular to a constant magnetic field B(vector)=BoZ-hat
a) What is the acceleration of the particle? Calculate the x, y and z-component of a as a function of time. You may choose the direction of the x and y-axes for your own convenience. Describe the particle's path as it moves under the influence of the magnetic field.
b) Find the power radiated per unit solid angle by the particle, assuming v << c.
c) Time average your result in part (b). (Just set up the expression for time average).
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Solution Summary
It investigates the motion of the charge in a constant magnetic field.
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Please see the attached file.
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Here is the plain TEX source
magnification=magstep1
baselineskip=12pt
parindent = 0pt
parskip = 12pt
defl{left}
defr{right}
defla{langle}
defra{rangle}
defp{partial}
centerline{bf Charged particle moving in a uniform magnetic field.}
As stated in part (b) we assume non-relativistic motion with $vll c$.
The ``Introduction to Electrodynamics" by David Griffiths will be referred to as
bf ``the book"rm.
bigskip
bf (a) rm
A particle with charge $q$ moving in a uniform magnetic field $vec B = B_ohat z$
is accelerated by the Lorentz force
$$
vec F = qvec v times vec B,
eqno(a.1)
$$
see equation (5.1) of the book.
Therefore its equation of motion (second law of Newton),
$$
vec a = vec F/m
eqno(a.2)
$$
where $m$ is its mass and $vec a$ is its acceleration,
can be written in Cartesian $(x,y,z)$ components as
$$
eqalign{
dot v_x &= {qB_oover m} v_y,cr
dot v_y &= -{qB_oover m} v_x,cr
dot v_z &= 0.cr
}
eqno(a.3)
$$
From the last of equations (a.3) we see that $v_z = const$ and the first two of
equations (a.3) can also be written as
$$
dot {vec v_bot} = -omega_o hat z times vec v_bot ,
eqno(a.4)
$$
where the Larmor frequency is
$$
omega_o = {qB_oover m}
eqno(a.5)
$$
and $vec v_bot = (v_x, v_y,0)$.
Equation (a.4) or equivalently the first two of equations (a.3)
describe a circular motion in ``$vec v$-space" with frequency $omega_o$ ...
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