4. Oscilloscope and Cyclotron

Oscilloscope:
An oscilloscope is an electronic device used to display and analyze the waveform of electronic signals. It uses the deflection of an electron beam in a vacuum tube by electric or magnetic fields to produce a visual representation of a signal.

Cyclotron:
A cyclotron is a type of particle accelerator that uses a combination of a constant magnetic field and a high-frequency alternating electric field to accelerate charged particles to high energies. The particles move in a spiral path due to the magnetic field, gaining energy each time they cross the electric field.

Working Principle of a Cyclotron:
The charged particles are injected into the cyclotron at the center.
A perpendicular magnetic field forces the particles into a circular path.
An alternating electric field between the two "dees" (semicircular electrodes) accelerates the particles each time they cross the gap between the dees.
As the particles gain energy, the radius of their path increases, causing them to spiral outward until they reach the desired energy and are extracted.

Example Problem:
Calculate the maximum energy of a proton accelerated by a cyclotron with a magnetic field of 1 T and a dee radius of 0.5 m.

Solution:
The kinetic energy K of the proton is given by:
K = (q² × B² × r²) / (2m)
where:
q is the charge of the proton,
B is the magnetic field strength,
r is the radius of the dee,
m is the mass of the proton.
Substituting the values:
K = [(1.6 × 10^-19)² × (1)² × (0.5)²] / [2 × 1.67 × 10^-27]
K = (6.4 × 10^-38) / (3.34 × 10^-27) = 1.92 × 10^-11 J
So, the maximum energy of the proton is 1.92 × 10^-11 J, or approximately 120 MeV (using 1 MeV = 1.6 × 10^-13 J).