Figure 1 shows a cyclotron. An alpha particle is released from rest and is accelerated each time it reaches the gap between two horizontal 'dees' D1D_1D1 and D2D_2D2. Between these accelerations the alpha particle moves at constant speed. A vertical magnetic field of flux density BBB acts over the dees so that the alpha particle follows a semicircular path in each dee.
Use the following constants in your calculations:
- Charge of an alpha particle, q=3.20×10−19 Cq = 3.20 \times 10^{-19}\text{ C}q=3.20×10−19 C
- Mass of an alpha particle, mα=6.64×10−27 kgm_{\alpha} = 6.64 \times 10^{-27}\text{ kg}mα=6.64×10−27 kg
- Fundamental charge unit, e=1.60×10−19 Ce = 1.60 \times 10^{-19}\text{ C}e=1.60×10−19 C
Explain why the alpha particle travels in a semicircular path of constant radius inside each dee.
The peak potential difference of the alternating supply is 25.0 kV25.0\text{ kV}25.0 kV. The alpha particle leaves the cyclotron with a kinetic energy of 24 MeV24\text{ MeV}24 MeV. Determine the number of times the alpha particle moves across the gap before it leaves the cyclotron.
The radius of the outermost semicircular path of the alpha particle is RRR and the alpha particle leaves with a maximum kinetic energy EkE_kEk. Show that EkE_kEk is given by: Ek=q2B2R22mαE_k = \frac{q^2 B^2 R^2}{2 m_{\alpha}}Ek=2mαq2B2R2
A medical research facility decides to purchase a cyclotron in order to manufacture its own radioactive isotopes using high-speed alpha particles.
- The required minimum kinetic energy of the emerging alpha particles is 20.0 MeV20.0\text{ MeV}20.0 MeV.
- The cost of a cyclotron is approximately proportional to Ek1.3E_k^{1.3}Ek1.3.
- The cost of a 10.0 MeV10.0\text{ MeV}10.0 MeV cyclotron is about £2.5 million.
Table 1 gives information for three cyclotrons X, Y, and Z.
Table 1:
| Cyclotron | B / TB\text{ / T}B / T | R / mR\text{ / m}R / m |
|---|---|---|
| X | 1.5 | 0.45 |
| Y | 1.2 | 0.65 |
| Z | 1.0 | 0.55 |
Deduce which cyclotron X, Y, or Z will satisfy the energy requirement for the lowest cost. Go on to determine the approximate cost of this cyclotron.