Calculate the cyclotron frequency for electrons in a B-field of \(0.2 \mathrm{~T}\).

When exploring the behavior of electrons under the influence of a magnetic field, it's essential to grasp what a magnetic field is. Simply put, a magnetic field is a vector field that surrounds magnets and electric currents, denoting the direction and strength of the magnetic force that can act on other currents and magnetic materials. Magnetic fields are measured in teslas (T) in the International System of Units (SI).

In the context of a cyclotron, a magnetic field is employed to bend the path of charged particles, like electrons. It ensures that the particles move in a circular trajectory within the cyclotron. When an electron with a negative charge moves through a magnetic field, it experiences a force perpendicular to both the direction of its motion and the direction of the magnetic field, leading to circular motion. This behavior is rooted in the Lorentz force, which governs the motion of charged particles in electromagnetic fields.

An electron carries a fundamental property known as charge, denoted as 'e'. This intrinsic property of matter is measured in coulombs (C). The charge of a single electron is approximately \( -1.6 \times 10^{-19} \) C, and it's this charge that interacts with a magnetic field to produce force and subsequent motion.

The negative sign signifies that the electron has a negative charge, as opposed to the positive charge carried by protons. It is precisely this charge that we factor into the equation for calculating the cyclotron frequency, which is the frequency at which an electron orbits in a magnetic field. The electron's charge, when exposed to a magnetic field, causes it to accelerate perpendicular to both the velocity of the electron and the magnetic field, leading to its spiral trajectory in a device like a cyclotron.

The mass of the electron is another crucial constant in calculating the cyclotron frequency. The mass of an electron is approximately \( 9.11 \times 10^{-31} \) kilograms. Though the mass of an electron is quite minuscule, it plays a significant role in determining its behavior in a magnetic field.

In our equation, the mass of the electron is the denominator, which means it's inversely proportional to the cyclotron frequency. In other words, the lighter the particle, the higher the frequency with which it will orbit in a magnetic field of a given strength. This is why electrons, being so light, can reach incredibly high speeds and frequencies in a cyclotron, allowing physicists and other scientists to study them in high-energy states.