Why does the particle spiral outward instead of staying in a circle of constant radius?

The particle gains kinetic energy each time it is accelerated by the electric field in the gap. In a uniform magnetic field, the radius of the circular path is proportional to momentum (r = mv/qB). As speed (v) increases, so does the radius, causing the particle to trace a larger semicircle each time it returns to a dee, resulting in the outward spiral.

What happens if the frequency of the oscillating electric field doesn't match the cyclotron frequency?

Acceleration becomes inefficient or stops. For optimal energy gain, the electric field must reverse polarity exactly when the particle arrives at the gap, so it is always pushed forward. If the frequencies are mismatched, the particle may encounter a decelerating field, lose energy, or simply not be accelerated consistently, breaking the resonant condition essential for the cyclotron's operation.

Why can't a cyclotron accelerate particles to arbitrarily high speeds?

This simulator uses classical (non-relativistic) physics, where the cyclotron frequency is constant. In reality, as particles approach a significant fraction of the speed of light, their relativistic mass increases. This changes their orbital frequency, causing them to fall out of sync with the fixed-frequency oscillating electric field, imposing a fundamental energy limit on simple cyclotrons.

What is the role of the magnetic field? Why can't we just use a strong electric field?

The magnetic field's sole purpose is to bend the particle's path into a closed loop, steering it back to the acceleration gap repeatedly. A linear accelerator uses only electric fields, but the particle passes each gap only once. The magnetic field in a cyclotron enables reuse of the same relatively small voltage gap many times, allowing a compact design to achieve high energies.

Cyclotron (Schematic)

A cyclotron is a particle accelerator that uses a combination of static magnetic and oscillating electric fields to propel charged particles to high energies along a spiral path. This simulator illustrates the core operating principle: a uniform magnetic field, directed perpendicularly out of the plane, causes a charged particle to move in a circular orbit due to the Lorentz force. The key relationship is the cyclotron frequency, ω_c = qB/m, which is independent of the particle's speed for non-relativistic velocities. This frequency determines the particle's orbital period. Between the two hollow 'dees,' an oscillating electric field is applied across the gap. Each time the particle crosses this gap, it is accelerated by the electric force, qE, provided the field's polarity is synchronized to reverse direction each half-cycle of the particle's motion. This synchronization causes the particle to gain kinetic energy with each crossing, increasing its speed and, consequently, the radius of its semicircular path within the dees, leading to the characteristic outward spiral. Students can explore the direct relationship between magnetic field strength and orbital frequency, observe the critical role of field synchronization, and see how energy gain translates to a larger spiral radius. The model simplifies real-world cyclotrons by neglecting relativistic mass increase (limiting it to classical regimes), assuming perfect vacuum and field uniformity, and using a schematic 2D representation. Interacting with this simulation reinforces understanding of the Lorentz force law (F = q(v × B)), uniform circular motion dynamics, and the ingenious method of resonant acceleration.

Who it's for: Undergraduate physics or engineering students studying electromagnetism, particularly the motion of charged particles in fields and the principles of particle accelerators.

Key terms

Cyclotron
Lorentz Force
Cyclotron Frequency
Particle Accelerator
Uniform Magnetic Field
Oscillating Electric Field
Synchronization
Spiral Path

How it works

Classical cyclotron idea: uniform B bends a charged particle into a spiral, while an alternating E in the gap between dees adds energy in step with cyclotron frequency ω_c = qB/m (non-relativistic). This canvas uses a central strip with E ∝ sin(ω_c t) and B out of the page — a pedagogical cartoon, not RF engineering or relativistic synchrotron corrections.

Key equations

F = q(E + v×B) · ω_c = qB/m · resonance: E flips each half-turn (ideal)

Frequently asked questions

Why does the particle spiral outward instead of staying in a circle of constant radius?: The particle gains kinetic energy each time it is accelerated by the electric field in the gap. In a uniform magnetic field, the radius of the circular path is proportional to momentum (r = mv/qB). As speed (v) increases, so does the radius, causing the particle to trace a larger semicircle each time it returns to a dee, resulting in the outward spiral.
What happens if the frequency of the oscillating electric field doesn't match the cyclotron frequency?: Acceleration becomes inefficient or stops. For optimal energy gain, the electric field must reverse polarity exactly when the particle arrives at the gap, so it is always pushed forward. If the frequencies are mismatched, the particle may encounter a decelerating field, lose energy, or simply not be accelerated consistently, breaking the resonant condition essential for the cyclotron's operation.
Why can't a cyclotron accelerate particles to arbitrarily high speeds?: This simulator uses classical (non-relativistic) physics, where the cyclotron frequency is constant. In reality, as particles approach a significant fraction of the speed of light, their relativistic mass increases. This changes their orbital frequency, causing them to fall out of sync with the fixed-frequency oscillating electric field, imposing a fundamental energy limit on simple cyclotrons.
What is the role of the magnetic field? Why can't we just use a strong electric field?: The magnetic field's sole purpose is to bend the particle's path into a closed loop, steering it back to the acceleration gap repeatedly. A linear accelerator uses only electric fields, but the particle passes each gap only once. The magnetic field in a cyclotron enables reuse of the same relatively small voltage gap many times, allowing a compact design to achieve high energies.

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