Chapter 2, Problem 2.22P

(a)

To determine

Show that the trajectory of a particle is $r=\frac{mv}{qB}=\frac{v}{{\omega }_c}$

(b)

To determine

Show that the z component of the motion is given by

    $z\left(t\right)=z_0+{\dot{z}}_{0}t+\frac{q{E}_z}{2m}{t}^2$

(c)

To determine

Show that the time averages of velocity components are

    $\begin{array}{l}\langle \dot{x}\rangle =\frac{E_y}{B}\\ \langle \dot{y}\rangle =0\end{array}$

(d)

To determine

Sketch the projection of the trajectory on the x-y plane for the given cases.