Protons are injected into the 1.0 -km-radius Fermilab...

Protons are injected into the 1.0 -km-radius Fermilab Tevatron with an energy of $150 \mathrm{GeV}$. If they are accelerated by $2.5 \mathrm{MV}$ each revolution, how far do they travel and approximately how long does it take for them to reach $1.0 \mathrm{TeV} ?$

Key Concepts

Particle Accelerators

Particle accelerators are devices used to accelerate charged particles to high speeds using electromagnetic fields. They often employ different methods—linear acceleration or circular motion—to impart energy to particles, enabling experiments in high-energy physics by colliding particles at near-light speeds.

Synchrotron Acceleration

Synchrotron acceleration is a process used in circular accelerators where particles are accelerated in discrete steps (or turns) by RF cavities that apply an energy boost each revolution. This method efficiently increases the particle’s energy over many cycles while keeping them on a closed-loop path defined by strong magnetic fields.

Circular Motion and Circumference

In a circular accelerator, the path length of the particles is determined by the circumference of the ring, which is calculated from the radius of the accelerator. Understanding the geometry of the accelerator is crucial for calculating the distance traveled per revolution and overall distance as the particle gains energy.

Energy Gain Per Revolution and Turn Counting

The concept of energy gain per revolution involves determining how many turns are required to reach a desired energy level by dividing the energy difference by the fixed energy gain provided each revolution. This quantization of energy gain is fundamental to planning the acceleration schedule in a synchrotron.

Relativistic Particle Dynamics

At energies approaching the TeV scale, charged particles like protons travel at speeds very near the speed of light. This relativistic behavior means that calculations regarding time and distance over many revolutions must take into account the nearly constant speed (approximately c) of the particles, even as their energy increases significantly.

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