Physics Problems & Solutions: Cyclotron

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Electromagnetism - Cyclotron

An electron in a metal has an effective mass m* = 0.1 m_e. If this metal is placed in a magnetic field of magnitude 1 tesla, the cyclotron resonance frequency ω_c, is most nearly

A. 930 rad/s
B. 8.5 × 10⁶ rad/s
C. 2.8 × 10¹¹ rad/s
D. 1.8 × 10¹² rad/s
E. 7.7 × 10²⁰ rad/s

(GR9277 #60)

Solution:

$F_\text{lorentz}=F_\text{centripetal}$

$qvB=\frac{mv^2}{r}$

Angular velocity: $\omega=2\pi f=\frac{v}{r}=\frac{qB}{m}$

$\begin{aligned}\omega_c&=\frac{q_eB}{m^*}\\&=\frac{1.6\times 10^{-19}\times 1}{0.1 m_e} \\&=\frac{1.6\times 10^{-19}}{0.1 \times9.11\times 10^{-31}}\\&=\frac{10.6\times 10^{-20}}{9.11\times 10^{-32}}\\&\approx 10^{12}\end{aligned}$

Answer: D

Special Relativity - Rest Mass

A particle leaving a cyclotron has a total relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass of this particle?

A. 0.25 GeV/c²
B. 1.20 GeV/c²
C. 2.00 GeV/c²
D. 6.00 GeV/c²
E. 16.0 GeV/c²

(GR0177 #79)

Solution:

E²= p²c²+ m₀²c⁴

Given: E = 10 GeV and p = 8 GeV/c

100= (64/c²)c²+ m₀²c⁴
m₀²c⁴= 100 − 64= 36
m₀= 6 GeV/c²

Answer: D

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