---
title: "Cyclotron protons, charge-to-mass ratio"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:cyclotron-protons-charge-mass-ratio"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/cyclotron-protons-charge-mass-ratio
site: BCT Engineering
---

# Cyclotron protons, charge-to-mass ratio

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2067 Asoj, 5 marks
- Video: [Physics 43  Magnetic Forces on Moving Charges (21 of 26) The Cyclotron](https://www.youtube.com/watch?v=GE07eLk2ojA) — Michel van Biezen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/cyclotron-protons-charge-mass-ratio
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

A cyclotron which has the dees of radius 42cm and magnetic field of flux density 0.5 weber/m² is employed to accelerate protons. If the final velocity of the proton is 2.02×10⁷ m/sec, calculate the charge to mass ratio for the proton and the frequency of the alternating potential between the dees.

## Short answer

Charge-to-mass ratio q/m = v/(BR) ≈ 9.62×10⁷ C/kg. Frequency f= v/(2πR) ≈ 7.65×10⁶ Hz = 7.65 MHz.

## Answer

At the final radius R (the dee radius) the proton moves with v = (q/m) B R, and its orbital frequency is f = qB/(2πm) = v/(2πR).

**Given:** R = 42 cm = 0.42 m, B = 0.5 Wb/m², v = 2.02×10⁷ m/s.

#### Charge-to-mass ratio
```text
q/m = v / (B R) = 2.02×10⁷ / (0.5 × 0.42)
    = 2.02×10⁷ / 0.21
    = 9.62×10⁷ C/kg
```
(The accepted value for the proton is 9.58×10⁷ C/kg.)

#### Frequency of the alternating potential
```text
f = (q/m) B / (2π) = (9.62×10⁷ × 0.5) / (2π)
  = 4.81×10⁷ / 6.283
  = 7.65×10⁶ Hz
(check: v/(2πR) = 2.02×10⁷ / (2π × 0.42) = 7.65×10⁶ Hz)
```

**Answer: q/m ≈ 9.6×10⁷ C/kg; f ≈ 7.65 MHz**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At the dee radius R the final speed is:
   A. v = (q/m)B/R
   B. v = (m/q)BR
   C. v = (q/m)BR
   D. v = qBR²/m
   Answer: C. From r = mv/(qB). (Key point: Speed relation)

2. Charge-to-mass ratio is:
   A. v/(BR) ≈ 9.62×10⁷ C/kg
   B. vBR ≈ 4.2×10⁶
   C. v/B ≈ 4×10⁷
   D. BR/v ≈ 1×10⁻⁸
   Answer: A. 2.02×10⁷/(0.5×0.42). (Key point: q/m)

3. The frequency of the alternating potential is:
   A. f = 2πqB/m ≈ 3×10⁸ Hz
   B. f = qB/(2πm) = v/(2πR) ≈ 7.65 MHz
   C. f = v R/(2π) ≈ 1.3×10⁶
   D. f = qBR/(2πm) at every orbit
   Answer: B. Both forms give 7.65×10⁶ Hz. (Key point: Frequency)

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