---
title: "Cyclotron working; semicircle time constant"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:cyclotron-time-semicircle-independent"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/cyclotron-time-semicircle-independent
site: BCT Engineering
---

# Cyclotron working; semicircle time constant

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2069 Chaitra, 5 marks; 2068 Chaitra, 5 marks (asked 2 times)
- Video: [Motion of Charge in a Magnetic Field - The Cyclotron | Physics with Professor Matt Anderson | M23-06](https://www.youtube.com/watch?v=HmVGGs6Dxp8) — Physics with Professor Matt Anderson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/cyclotron-time-semicircle-independent
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Describe the principle and working of Cyclotron. Show that the time taken by the ion in a Dee to travel a semicircle is exactly the same whatever be its radius and velocity.

## Short answer

A cyclotron uses a magnetic field to bend ions into semicircles and an oscillating electric field to accelerate them. The time for a semicircle is t = πm/(qB), independent of radius and speed, so a fixed oscillator frequency qB/(2πm) maintains resonance.

## Answer

#### Principle
A charged particle moving at right angles to a uniform magnetic field travels on a circle, and its period of revolution does not depend on its speed. A small oscillating electric field of the same frequency, applied across a gap, gives the particle an energy kick every half turn, so it spirals outward to very high energy without needing a high voltage.

#### Construction
```text
        B (into page, uniform)
       ______   ______
      /      \ /      \
     |   D1   | |  D2   |
     |    ~ ~ ~~~ ~ ~   |   <- gap with RF
     |  ion   |gap|     |      voltage
      \______/ \______/
          ~ oscillator ~
```
Two hollow D-shaped metal boxes (dees) in an evacuated chamber between the poles of an electromagnet. An ion source is at the centre; a high-frequency oscillator is connected across the dees.

#### Working
1. An ion (charge q, mass m) at the centre is attracted to the dee of opposite polarity and accelerates across the gap.
2. Inside the dee there is no electric field. The magnetic field bends it into a semicircle.
3. When it returns to the gap, the oscillator has reversed polarity, so the ion is accelerated again, gaining energy qV.
4. The radius increases each time; after many turns the ion reaches the rim and leaves through a deflector with energy K = q²B²R²/(2m).

#### Time for a semicircle
1. Magnetic force provides the centripetal force: qvB = mv²/r
2. r = mv/(qB), so the time for a semicircle is

t = πr/v = π (mv/qB)/v = **πm/(qB)**

This has no r or v in it. Hence the time is the same for every semicircle, whatever the radius and speed, so a fixed frequency f = qB/(2πm) keeps the ion in step with the oscillator (the resonance condition).

## Check yourself

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

1. The cyclotron works because the period of revolution
   A. Depends on speed
   B. Is independent of speed
   C. Depends on radius
   D. Is zero
   Answer: B. With fixed frequency, the oscillator stays in step. (Key point: Principle)

2. Inside a dee the ion
   A. Is accelerated
   B. Feels no electric field and moves in a semicircle
   C. Stops
   D. Moves straight
   Answer: B. The dee shields electric field; the magnetic field bends it. (Key point: Inside the dee)

3. The time for a semicircle is
   A. πm/(qB)
   B. 2πm/(qB)
   C. πmv/(qB)
   D. m/(qB)
   Answer: A. t = πr/v with r = mv/qB gives πm/(qB). (Key point: Time)

4. The resonance frequency is
   A. qB/(πm)
   B. qB/(2πm)
   C. 2πm/(qB)
   D. qBR/m
   Answer: B. f = 1/T = qB/(2πm). (Key point: Resonance)

## Related in Engineering Physics

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