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Homework Help: Relativistic form of Newton's second law

  1. Jan 24, 2010 #1
    SOLVED
    (Problem 10, Chapter 2, Modern Physics - Serway)
    1. The problem statement, all variables and given/known data
    Recall that the magnetic force on a charge q moving with velocity [tex]\vec{v}[/tex] in a magnetic field [tex]\vec{B}[/tex] is equal to [tex]q\vec{v}\times\vec{B}[/tex]. If a charged particle moves in a circular orbit with a fixed speed [tex]v[/tex] in the presence of a constant magnetic field, use the relativistic form of Newton's second law to show that the frequency of its orbital motion is

    [tex]
    f=\frac{qB}{2\pi m}(1-\frac{v^2}{c^2})^{1/2}
    [/tex]

    2. Relevant equations

    [tex]
    F=\frac{ma}{(1-v^2/c^2)^{3/2}}
    [/tex]

    3. The attempt at a solution
    The particle moves in a circle then the magnetic field is perpendicular to the velocity and [tex]F=qvB[/tex].

    [tex]
    f=\frac{v}{2\pi R}
    [/tex]

    [tex]
    qvB=\frac{ma}{(1-v^2/c^2)^{3/2}}
    =\frac{m}{(1-v^2/c^2)^{3/2}}\frac{v^2}{R}
    [/tex]

    [tex]
    R=\frac{mv}{(1-v^2/c^2)^{3/2}qB}
    [/tex]

    [tex]
    f=\frac{(1-v^2/c^2)^{3/2}qB}{2\pi m}
    [/tex]

    What's wrong?
     
    Last edited: Jan 24, 2010
  2. jcsd
  3. Jan 24, 2010 #2
    The relativistic form of Newton's second law is

    [tex]
    \frac{\partial\vec{p}}{\partial t} = \vec{F},
    [/tex]

    where

    [tex]
    \vec{p} = m\gamma(v)\vec{v}.
    [/tex]

    The factor

    [tex]
    \gamma(v) = \frac{1}{\sqrt{1-(v/c)^2}}
    [/tex]

    is constant in this task. Do not differentiate it!
    Write the differential equation system for [tex]v_x[/tex] and [tex]v_y[/tex] and derive the frequency.

    Good luck!
     
  4. Jan 24, 2010 #3
    Thanks!
     
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