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Exercises

  1. An electron of mass $m$ and charge $-e$ moves in a uniform $y$-directed electric field of magnitude $E$, and a uniform $z$-directed magnetic field of magnitude $B$. The electron is situated at the origin at $t=0$ with an initial $x$-directed velocity of magnitude $v_0 $. Show that the electron traces out a cycloid of the general form
    $\displaystyle x$ $\textstyle =$ $\displaystyle a\,\sin(\omega\,t) + b\,t,$  
    $\displaystyle y$ $\textstyle =$ $\displaystyle c\,[1-\cos(\omega\,t)],$  
    $\displaystyle z$ $\textstyle =$ $\displaystyle 0.$  

    Find the values of $a$, $b$, $c$, and $\omega$, and sketch the electron's trajectory in the $x$-$y$ plane when $v_0<E/B$, $E/B < v_0 < 2\,E/B$, and $v_0> 2\,E/B$.

  2. A particle of mass $m$ and charge $q$ moves in the $x$-$y$ plane under the influence of a constant amplitude rotating electric field which is such that $E_x= E_0\,\cos(\omega\,t)$ and $E_y=E_0\,\sin(\omega\,t)$. The particle starts at rest from the origin. Determine its subsequent motion. What shape is the particle's trajectory?

  3. A particle of mass $m$ slides on a frictionless surface whose height is a function of $x$ only: i.e., $z=z(x)$. The function $z(x)$ is specified by the parametric equations
    $\displaystyle x$ $\textstyle =$ $\displaystyle A\,[2\,\phi + \sin (2\,\phi)],$  
    $\displaystyle z$ $\textstyle =$ $\displaystyle A\,[1-\cos (2\,\phi)],$  

    where $\phi$ is the parameter. Show that the total energy of the particle can be written

    \begin{displaymath}
E = \frac{m}{2}\left(\frac{ds}{dt}\right)^{2} + \frac{1}{2}\,\frac{m\,g}{4\,A}\,s^2,
\end{displaymath}

    where $s=4\,A\,\sin\phi$. Deduce that the particle undergoes periodic motion whose frequency is amplitude independent (even when the amplitude is large). Demonstrate that the frequency of the motion is given by $4\pi\,(A/g)^{1/2}$.

next up previous
Next: Planetary Motion Up: Multi-Dimensional Motion Previous: Charged Particle Motion in
Richard Fitzpatrick 2011-03-31