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
   

   $$x = a\,\sin(\omega\,t) + b\,t,$$

   $$y = c\,[1-\cos(\omega\,t)],$$

   $$z = 0.$$

   
   
   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
   

   $$x = A\,[2\,\phi + \sin (2\,\phi)],$$

   $$z = A\,[1-\cos (2\,\phi)],$$

   
   
   where $\phi$ is the parameter. Show that the total energy of the particle can be written
   
   $$E = \frac{m}{2}\left(\frac{ds}{dt}\right)^{2} + \frac{1}{2}\,\frac{m\,g}{4\,A}\,s^2,$$
   
   
   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 $4\pi\,(A/g)^{1/2}$.