Cyclotron - UNSOLVED PRACTICE SET

Q7. Explain the principle of a cyclotron. Why is it called a "resonance accelerator"?

Q8. Why must the frequency of the oscillator in a cyclotron match the cyclotron frequency of the particle?

Q9. A cyclotron has magnetic field B = 1 T and dee radius R = 50 cm. Calculate the maximum kinetic energy of protons that can be accelerated.

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q10. Why can't a simple cyclotron accelerate electrons to very high energies? What is the relativistic limitation?

Q11. In a cyclotron, the potential difference between the dees is V. Does the energy gained per revolution depend on V? Explain.

Q12. Why are the dees in a cyclotron made hollow and shaped like a "D"?

Q13. Explain the construction and working of a cyclotron with a neat diagram. Show how the electric field accelerates the particle while the magnetic field keeps it in a circular path. Why must the oscillator frequency be constant?

Q14. Derive the expression for the maximum kinetic energy of a particle accelerated in a cyclotron. Show that it depends on the square of the dee radius and the square of the magnetic field.

Q15. A cyclotron is used to accelerate alpha particles (He²⁺ ions). The magnetic field is B = 1.2 T and the dee radius is R = 60 cm.

(a) Calculate the cyclotron frequency.  

(b) Calculate the maximum kinetic energy in MeV.  

(c) Calculate the number of revolutions needed to reach maximum energy if the peak voltage between dees is 50 kV.  

(d) Calculate the time taken to reach maximum energy.

[Given: m_α = 6.64 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q16. In a school science exhibition, a student builds a model cyclotron using a small electromagnet. The model has dee radius R = 10 cm, magnetic field B = 0.5 T, and oscillator frequency f = 7.6 MHz.

(a) Verify that the oscillator frequency matches the cyclotron frequency for protons.  

(b) Calculate the maximum speed and kinetic energy of protons.  

(c) If the peak voltage between dees is 500 V, how many revolutions are needed to reach maximum energy?  

(d) Calculate the time taken and the total distance traveled by the proton.  

(e) A visitor asks why the model can't accelerate electrons. Give a clear explanation.

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q17. A medical cyclotron produces radioactive isotopes for PET scans. It accelerates protons to 16 MeV using a magnetic field B = 1.5 T.

(a) Calculate the radius of the cyclotron dees.  

(b) Calculate the cyclotron frequency.  

(c) If the beam current is 100 μA, how many protons strike the target per second?  

(d) The target material is oxygen-18. Each proton reaction produces one fluorine-18 atom (half-life 110 minutes). If the irradiation lasts 30 minutes, what fraction of the maximum possible fluorine-18 is produced?  

(e) Explain why cyclotron-produced isotopes are preferred over reactor-produced isotopes for medical imaging.

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q18. A synchrocyclotron is a modified cyclotron where the oscillator frequency is gradually decreased to account for relativistic mass increase.

(a) Explain why the simple cyclotron fails at relativistic energies. Derive the condition showing when relativistic effects become significant.  

(b) In a synchrocyclotron, the frequency is modulated as f = f₀/(1 + K.E./mc²). For protons accelerated to 400 MeV, calculate the percentage change in frequency from start to finish.  

(c) If the dee radius is 1 m and B = 1.8 T, compare the maximum non-relativistic energy with the actual relativistic energy achievable.  

(d) Explain the role of the "frequency modulation" in maintaining resonance.  

(e) Why are modern particle accelerators (like LHC) based on synchrotrons rather than cyclotrons?

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C, c = 3 × 10⁸ m/s]