Cyclotron Radius Calculator






Introduction

A charged particle moving through a magnetic field does not travel in a straight line. It curves into a circular path, and the radius of that circle is called the cyclotron radius. If you know the particle's mass, its speed, its charge and the strength of the magnetic field, you can work out that radius in seconds with the calculator above.

What Is Cyclotron Radius?

Cyclotron radius is the radius of the circular path followed by a charged particle moving at right angles to a uniform magnetic field. It is also called the gyroradius or Larmor radius. The stronger the magnetic field, or the larger the charge, the tighter the circle. A heavier or faster particle, on the other hand, makes a wider circle.

Cyclotron Radius Formula

r = mv / (|q| × B)

Where:

  • r = cyclotron radius, in metres (m)
  • m = particle mass, in kilograms (kg)
  • v = particle velocity, in metres per second (m/s)
  • q = particle charge, in coulombs (C) — only the size of the charge is used, so the sign is ignored
  • B = magnetic field strength, in teslas (T)

The formula only holds when the particle moves at right angles to the field. If the motion is at an angle, only the part of the velocity at right angles to the field sets the radius.

Worked Example 1

A particle has mass 2 kg, velocity 3 m/s, charge 1 C and moves in a magnetic field of 2 T.

r = (2 × 3) / (1 × 2) = 6 / 2 = 3 m.

So the cyclotron radius is 3 metres.

Worked Example 2

A particle has mass 4 kg, velocity 5 m/s, charge 2 C and moves in a magnetic field of 2 T.

r = (4 × 5) / (2 × 2) = 20 / 4 = 5 m.

So the cyclotron radius is 5 metres.

How to Use the Calculator

  1. Enter the particle mass in kilograms.
  2. Enter the particle velocity in metres per second.
  3. Enter the charge in coulombs.
  4. Enter the magnetic field strength in teslas.
  5. Click Calculate. The cyclotron radius appears in metres below the button.

Common Mistakes to Avoid

  • Mixing up units: mass must be in kilograms, not grams, and velocity in metres per second, not kilometres per hour.
  • Using a negative charge value: the formula uses the size of the charge, so enter the charge as a positive number.
  • Forgetting that the formula assumes motion at right angles to the field. For angled motion, the radius comes from the perpendicular part of the velocity only.

Where Cyclotron Radius Matters

Cyclotron radius matters wherever charged particles move in magnetic fields: in cyclotron particle accelerators, in magnetic confinement for fusion research, in mass spectrometers that separate particles by mass and charge, and in space physics, where charged particles spiral along Earth's magnetic field.

Related Calculators

Frequently Asked Questions

  1. What is cyclotron radius? — The radius of the circular path a charged particle follows in a magnetic field.
  2. What is the formula for cyclotron radius? — r = mv / (|q|B), where m is mass, v is velocity, q is charge and B is magnetic field strength.
  3. What unit is cyclotron radius measured in? — Metres (m), when mass is in kilograms, velocity in metres per second, charge in coulombs and field strength in teslas.
  4. Does a stronger magnetic field increase or decrease cyclotron radius? — Decrease. A stronger field bends the particle more tightly, so the circle gets smaller.
  5. Does charge sign matter in the formula? — No. Only the size of the charge is used, so a negative charge gives the same radius as a positive charge of the same size.
  6. What happens if the particle moves parallel to the magnetic field? — It feels no magnetic force from that field and does not curve, so this formula does not apply.

Conclusion

Cyclotron radius is a simple ratio: momentum on top, charge times field strength below. Enter your four values in the calculator above, check that your units match, and you get the radius in metres straight away.

Similar Posts