To calculate the maximum energy of a cosmic-ray proton, you can use the gyroradius formula with the limits of the spherical source radius and magnetic field strength, incorporating the mass and charge of the proton.
The maximum energy of cosmic-ray protons depends on the gyroradius being smaller than the size of the acceleration region. Given a spherical source radius of R = 100 parsec and magnetic field strength of B = 1 micro-gauss, one can use the equation provided in pertinent lectures to find this maximum energy. Applying the gyro-radius formula r = mv/(qB) under the condition that r < R and solving for the energy E = 1/2 mv², one arrives at the energy value dependent on B, R, and fundamental constants of a proton's mass (m) and charge (q).
The Lorentz force acting perpendicular to the motion and the magnetic field dictates the path of the particle. Assuming full efficiency in the conversion of potential acceleration to energy, and using natural units where the speed of light c = 1, the energy (E) can be represented in electron volts (eV) as it is a common unit for energy in particle physics.