1. What is a Cyclotron Frequency Calculator?

Definition: This calculator computes the cyclotron frequency, which is the frequency at which a charged particle orbits in a uniform magnetic field due to the balance of the Lorentz force and centrifugal force.

Purpose: It is used in physics and engineering to design cyclotrons, analyze particle motion in magnetic fields, and understand phenomena in plasma physics and astrophysics.

2. How Does the Calculator Work?

The cyclotron frequency is calculated using:

Where:

• \( f \) is the cyclotron frequency (Hz)
• \( q \) is the charge of the particle (C)
• \( B \) is the magnetic field strength (T)
• \( m \) is the mass of the particle (kg)

Steps:

• Enter the charge (\( q \)) and select a unit (pC, nC, µC, mC, C)
• Enter the magnetic field (\( B \)) and select a unit (mT, T)
• Enter the mass (\( m \)) and select a unit (pg, ng, µg, mg, g, kg)
• Convert charge, magnetic field, and mass to C, T, and kg, respectively
• Calculate the frequency using the formula
• Convert the frequency to Hz, kHz, and MHz

Display format:

• If a frequency value is greater than 10000 or less than 0.0001 in absolute terms (and not zero), it is displayed in scientific notation (e.g., \( 1.234567e-12 \))
• Otherwise, it is displayed in decimal format with 5 decimal places

3. Importance of Cyclotron Frequency Calculation

Details: The cyclotron frequency is fundamental in particle accelerators like cyclotrons, used in medical imaging (e.g., PET scans), and in studying charged particle behavior in magnetic fields, such as in the Earth’s magnetosphere or in fusion research.

4. Using the Calculator

Tips: Enter the charge (\( q \)), magnetic field (\( B \)), and mass (\( m \)) with their units. The calculator will compute the cyclotron frequency in Hz, kHz, and MHz, with 5 decimal places. If the frequency is greater than 10000 or less than 0.0001 in absolute terms, it will be displayed in scientific notation.

Examples:

• Electron in a magnetic field: \( q = 1.602 \times 10^{-19} \, \text{C} \), \( B = 1 \, \text{T} \), \( m = 9.109 \times 10^{-31} \, \text{kg} \):
  • \( f = \frac{(1.602 \times 10^{-19}) \times 1}{2\pi \times 9.109 \times 10^{-31}} = 2.79924e+10 \, \text{Hz} \)
  • kHz = \( 2.79924e+07 \), MHz = \( 27992.40000 \)
• Proton in a cyclotron: \( q = 1.602 \times 10^{-19} \, \text{C} \), \( B = 0.5 \, \text{T} \), \( m = 1.673 \times 10^{-27} \, \text{kg} \):
  • \( f = \frac{(1.602 \times 10^{-19}) \times 0.5}{2\pi \times 1.673 \times 10^{-27}} = 7.62142e+06 \, \text{Hz} \)
  • kHz = \( 7621.42000 \), MHz = \( 7.62142 \)

5. Frequently Asked Questions (FAQ)

Q: What is cyclotron frequency?
A: It’s the frequency at which a charged particle orbits in a uniform magnetic field, resulting from the balance of the Lorentz force and centrifugal force.

Q: Why does the equation not account for relativistic effects?
A: The formula assumes non-relativistic speeds. At high speeds, relativistic effects increase the particle’s effective mass, reducing the frequency.

Q: Why are some frequency values displayed in scientific notation?
A: Values greater than 10000 or less than 0.0001 in absolute terms (and not zero) are displayed in scientific notation for clarity.

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