The frequency of revolution of the second particle is twice the frequency of revolution of the first particle.
The frequency of revolution of a charged particle in a uniform magnetic field is given by:
f = qB / 2πm
where:
f is the frequency of revolution (in revolutions per second)
q is the charge of the particle (in coulombs)
B is the strength of the magnetic field (in teslas)
m is the mass of the particle (in kilograms)
Since both particles are moving perpendicular to the same magnetic field, the magnetic field strength B is the same for both particles.
Additionally, the second particle has the same charge sign as the first particle, so the force on the second particle is in the same direction as the force on the first particle.
This means that the second particle will also move in a circular orbit with the same radius as the first particle.
The only difference between the two particles is that the second particle has twice the charge of the first particle.
This means that the force on the second particle is twice as strong as the force on the first particle.
As a result, the second particle will have a greater centripetal acceleration than the first particle.
The centripetal acceleration of a particle moving in a circular orbit is given by:
a = v²/r
where:
a is the centripetal acceleration (in meters per second squared)
v is the speed of the particle (in meters per second)
r is the radius of the orbit (in meters)
Since the second particle has a greater centripetal acceleration than the first particle, it will have a shorter period of revolution. The period of revolution is the time it takes for the particle to complete one full revolution of its orbit. It is given by:
T = 1/f
where:
T is the period of revolution (in seconds)
f is the frequency of revolution (in revolutions per second)
Since the second particle has a shorter period of revolution, it will have a higher frequency of revolution than the first particle.
The frequency of revolution of the second particle is given by: f = 2f.