1 \begin{problem*}{23.53} % cyclotrons
2 A particle with a mass of $m = 2.00\E{-16}\U{kg}$ and a charge of $q =
3 30.0\U{nC}$ starts from rest, is accelerated by a strong electric
4 field, and is fired from a small source inside a region of uniform
5 constant magnetic field $B = 0.600\U{T}$. The velocity of the
6 particle is perpendicular to the field. The circular orbit of the
7 particle encloses a magnetic flux of $\Phi_B = 15.0\U{$\mu$Wb}$.
8 \Part{a} Calculate the speed of the particle.
9 \Part{b} Calculate the potential difference through which the particle
10 accelerated inside the source.
15 For particles circling in a uniform, perpendicular magnetic field,
17 F_c &= m \frac{v^2}{r} = qvB \\
21 Letting $\tau$ be the period, from $\Delta x = v \Delta t$ we have
23 \tau = \frac{2 \pi r}{v} = \frac{2 \pi r m}{qrB} = \frac{2 \pi m}{qB}
26 The inverse of our cyclotron frequency from Recitation 7.
28 The flux and magnetic field give us radius by
30 \Phi_B &= AB = \pi r^2 B \\
31 r &= \sqrt{\frac{\Phi_B}{\pi B}} = \ans{2.82\U{mm}}
34 So the speed is given by
36 v = \frac{2 \pi r}{\tau} = \frac{qB}{2 \pi m}\sqrt{\frac{\Phi_B}{\pi B}}
43 K &= \frac{1}{2}mv^2 = q\Delta V \\
44 \Delta V &= \frac{mv^2}{2q} = \ans{215\U{V}}