Cyclotron frequency | Moving charges & magnetism | Physics | Khan Academy

Summary

This script explores the functioning of cyclotrons, devices that accelerate charged particles using oscillating electric and magnetic fields. It delves into the crucial timing of the electric field's oscillation, which must match the particle's spiraling motion to maintain constant acceleration. The script explains the calculation of the cyclotron's frequency, emphasizing the importance of a constant frequency for proper operation. It also touches on the relativistic effects that become significant at near-light speeds, which challenge the cyclotron's design by altering the particle's mass and thus the required oscillator frequency, leading to the development of synchrotrons for high-speed particle acceleration.

Takeaways

• 🔋 Cyclotrons use oscillating electric fields to accelerate charged particles, with magnetic fields helping to turn the particles and re-enter the electric field for continuous acceleration.
• 🕒 The electric field in a cyclotron must flip at the right moment to ensure the particle is accelerated at each turn, which is synchronized with the particle's spiral motion.
• 🔄 The frequency of the electric field flipping should remain constant because the time taken to complete one spiral remains the same, despite the increasing radius due to increasing speed.
• ⚙️ The time for one oscillation of the electric field is equal to the time for one spiral of the particle, which is a crucial synchronization for the cyclotron's operation.
• 📉 The time taken to complete half a circle (or one spiral) does not change as the particle accelerates, because the increase in distance is proportionate to the increase in speed.
• 🔢 The time for one full spiral can be calculated using the formula \( \frac{2\pi m}{qB} \), where \( m \) is the mass of the particle, \( q \) is the charge, and \( B \) is the magnetic field strength.
• 🔡 The frequency of the oscillator in a cyclotron, often called the cyclotron frequency, is the reciprocal of the time for one full spiral and is given by \( \frac{qB}{2\pi m} \).
• 🌀 The spiral motion of particles in a cyclotron is a result of the balance between the Lorentz force due to the magnetic field and the centripetal force required for circular motion.
• 🚀 Cyclotrons are limited by Einstein's theory of relativity, as the increasing mass of particles at speeds close to the speed of light affects the constant frequency of the oscillator.
• 🔄 Modern accelerators, such as synchrotrons, can adjust the oscillator frequency to accommodate relativistic effects and allow particles to reach speeds closer to the speed of light.
• 🛠 Understanding the principles of cyclotrons is fundamental to grasping the operation of more advanced particle accelerators and their limitations.

Q & A

What is the basic principle behind the operation of a cyclotron?

A cyclotron accelerates charged particles using oscillating electric fields. As the particles move, a magnetic field causes them to turn and re-enter the electric field, allowing for continuous acceleration in a spiral path until they reach the maximum radius and are ejected at high speed.

Why is it important for the electric field in a cyclotron to flip at the right moment?

The electric field must flip at the right moment to ensure that the charged particles are continuously accelerated. If the field does not flip in sync with the particle's motion, the particles will not gain the necessary speed to continue their spiral path and reach the maximum radius.

Should the frequency of the electric field flipping in a cyclotron be constant or changing?

The frequency should remain constant. This is because the time it takes for a particle to complete one spiral remains the same as it spirals outwards, due to the proportional increase in both the distance (radius) and the speed of the particle.

How can the frequency at which the electric field should flip be calculated for a cyclotron?

The frequency can be calculated using the formula for the cyclotron frequency, which is the reciprocal of the time period of one full spiral. The time period is given by "T = 2πm/qB", where "m" is the mass of the particle, "q" is the charge, and "B" is the magnetic field strength. The frequency "f" is then "f = qB/2πm".

What is the significance of the time it takes for a proton to complete one spiral in a cyclotron?

The time it takes for a proton to complete one spiral is crucial because it determines the time period of the oscillator. This time period must be constant for the cyclotron to function properly, as it dictates the frequency at which the electric field must flip to keep accelerating the proton.

Why does the time taken to complete a half-circle or a full circle in a cyclotron remain constant?

The time remains constant because the increase in the path length (circumference) as the proton spirals outwards is proportionate to the increase in its speed. This means that the time taken, which is the ratio of distance to speed, does not change.

What is the major limitation of cyclotrons when it comes to accelerating particles to very high speeds?

The major limitation is the effect of Einstein's theory of relativity, where mass increases with velocity as particles approach the speed of light. This increase in mass affects the required oscillator frequency, which must be adjusted if particles are to be accelerated to speeds close to the speed of light.

How do modern accelerators overcome the limitations of cyclotrons at high speeds?

Modern accelerators, such as synchrotrons or synchro-cyclotrons, adjust the frequency of the oscillator as the particles spiral out. This ensures that the acceleration process remains in sync with the increasing speed of the particles.

What is the relationship between the mass of a particle, its charge, the magnetic field, and the time it takes to complete a spiral in a cyclotron?

The time it takes to complete a spiral is directly proportional to the mass of the particle and inversely proportional to the product of its charge and the magnetic field strength. This relationship is constant for a given cyclotron setup, ensuring a constant time period for the oscillator.

Why is the speed of a particle in a cyclotron proportional to the radius of its path?

The speed of a particle is proportional to the radius because the magnetic force acting as the centripetal force is given by "F = qvB", where "v" is the speed and "r" is the radius. Rearranging this formula shows that "v = qBr/m", indicating that speed increases with radius as the particle spirals outwards.